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ts inter trigonometric equations 4 marks important questions 2024

TS Inter Trigonometric Equations – 4-M Important Questions

Education, TS Inter Maths, TS Inter Maths 1A / Nayini Satyanarayana Reddy

Trigonometric Equations – 4-Marks important Questions Trigonometric Equations “Trigonometric Equations 4-Mark Important Questions” would typically refer to a collection of questions worth four marks each that focus on solving equations involving trigonometric functions. These questions are likely intended for students studying trigonometry at the intermediate level or equivalent. Trigonometric equations involve expressions containing trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. The goal in solving these equations is typically to find the values of the variable(s) that satisfy the given equation within a specified interval. Trigonometric Equations These types of questions may cover various topics within trigonometric equations, including: Solution of basic trigonometric equations: These equations involve single trigonometric functions and can often be solved using algebraic techniques such as factoring, substitution, or trigonometric identities. Trigonometric equations involving multiple angles: Equations may involve multiple angles, such as sums, differences, or multiples of trigonometric functions. Students may need to apply trigonometric identities or properties to simplify the equations before solving them. Trigonometric Equations Trigonometric Equations Students may encounter equations where trigonometric identities need to be applied to rewrite the equation in a more simplified form before solving it. Solutions of Trigonometric Equations with restrictions: Some equations may have restrictions on the domain, such as finding solutions within a specific interval or range of values. Solving Trigonometric Equations involving transformations: Equations may involve transformations of trigonometric functions, such as amplitude changes, phase shifts, or vertical and horizontal translations. SAQs of Trigonometric Equations serve as a means for students to practice and demonstrate their understanding of trigonometric equations, their ability to apply various problem-solving techniques, and their proficiency in manipulating trigonometric functions to find solutions. Trigonometric Equations Additionally, these questions may also help students prepare for assessments or examinations where solving trigonometric equations is a key component of the curriculum. Visit my YouTube channel: Click on the Logo

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ts inter Trigonometric Ratios Up To Transformations 4 marks important questions

TS Inter Trigonometric Ratios up to Transformations – 4-Mark Questions

Education, TS Inter Maths, TS Inter Maths 1A / Nayini Satyanarayana Reddy

TS Inter Trigonometric Ratios up to Transformations – 4-Mark Questions This comprehensive guide is designed to assist students in mastering the fundamental concepts of trigonometric ratios, with a focus on understanding transformations. Tailored to ensure a thorough understanding of the topic, this resource highlights essential questions worth 4 marks each, providing targeted practice for examinations. Through clear explanations and strategic problem-solving techniques, students will gain confidence in manipulating trigonometric functions within various transformations, paving the way for success in both classroom assessments and standardized tests. “Trigonometric Ratios Up To Transformations: 4 Marks Important Questions” is a meticulously crafted resource aimed at sharpening students’ understanding of trigonometry, specifically focusing on transformations. Within its pages, learners will encounter a curated selection of questions, each worth 4 marks, strategically chosen to reinforce key concepts and test problem-solving skills. TS inter Maths 1A Question Papers Trigonometric ratios upto transformations, maths 1a trigonometric ratios upto transformations important, trigonometric ratios, the product of vectors important questions for ipe, trigonometry 4 marks important questions, trigonometric functions 5marks important questions, most important 4 marks questions,4 marks most important questions in Telugu, most important 4 marks questions 1a and 1b, most important 4 marks questions in Telugu, trigonometric functions important questions 202 Trigonometric Ratios Up To Transformations: 4 Marks Important Questions” is a comprehensive study companion meticulously crafted to aid students in conquering trigonometry. Delving into the pivotal realm of transformations, this resource presents a curated selection of questions, each strategically chosen to reinforce key concepts and hone problem-solving abilities. With clear explanations and step-by-step solutions, learners can navigate through the complexities of trigonometric functions within transformations with confidence. Whether preparing for exams or seeking to deepen comprehension, this guide is an invaluable tool for achieving mastery in trigonometry. These types of questions are likely aimed at testing students’ understanding of trigonometric concepts, their ability to apply these concepts to solve problems, and their proficiency in handling transformations of trigonometric functions. “TS Inter Trigonometric Ratios up to Transformations – 4-Mark Questions” sounds like a resource or a section within a textbook or study material designed for students studying trigonometry in the Telangana State Intermediate education system. This section likely contains questions that are worth four marks each, focusing on trigonometric ratios and their applications, including transformations. In trigonometry, understanding trigonometric ratios like sine, cosine, and tangent, and how they relate to angles in a right triangle or on the unit circle, is fundamental. Transformations in trigonometry may include changes in amplitude, frequency, phase shift, and vertical or horizontal translations of trigonometric functions. Visit my Youtube Channel: Click on Below Logo

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how to prepare ts inter maths examination

How to Prepare well for the TS Inter Maths Examination 1

Education, TS Inter Maths, TS Inter Maths 1A / Nayini Satyanarayana Reddy

How to Prepare well for the TS Inter Maths Examination How to Prepare: Preparing for the TS Inter WellMaths Examination requires a combination of effective study strategies and time management. Here are some tips to help you: Understand the Syllabus: Familiarize yourself with the entire syllabus. Identify the topics that carry more weight and focus on them first. Create a Study Schedule: Plan your study sessions. Allocate dedicated time for each topic, ensuring you cover the entire syllabus before the exam. Practice Regularly: Mathematics requires consistent practice. Solve a variety of problems, including those from previous years’ question papers. This helps you understand the exam pattern and boosts your confidence. Conceptual Clarity: Ensure a strong understanding of the fundamental concepts. If you encounter difficulties, seek help from your teachers, classmates, or online resources. Make Notes: Prepare concise notes for each chapter. These notes can serve as a quick revision tool before the exam. Use Reference Books: Refer to additional study materials and reference books to gain a deeper insight into complex topics. Choose books that align with your syllabus. Mock Tests: Take mock tests regularly to simulate exam conditions. This helps improve your time management skills and identifies areas that need further attention. Clarify Doubts: Don’t hesitate to clarify doubts with your teachers or classmates. Understanding every concept thoroughly is crucial for success in mathematics. Healthy Lifestyle: Ensure a balance between study and relaxation. Get adequate sleep, maintain a healthy diet, and take short breaks during study sessions to stay focused. Revision: Regularly revise what you’ve studied. Focus on the topics where you feel less confident. Repetition helps reinforce the concepts in your memory. Stay Positive: Maintain a positive mindset. Believe in your abilities and stay confident. Avoid last-minute stress, and trust the efforts you’ve put into your preparation. Remember, consistent and organized preparation is key. Good luck with your TS Inter Maths Examination! Group Study: How to Prepare Collaborate with classmates for group study sessions. Discussing concepts with others can provide different perspectives and deepen your understanding. Use Technology: Leverage educational apps, online resources, and interactive tutorials to supplement your learning. Many platforms offer practice quizzes and video lessons. Stay Organized: Keep your study materials and notes organized. A well-structured study environment can help you focus better and save time when revisiting topics. Time Management: Practice time management during your study sessions and exams. Allocate specific time limits for each question while solving practice papers to improve efficiency. Focus on Weak Areas: Identify your weaker areas and allocate more time to them. It’s essential to address your weaknesses rather than avoiding them. Stay Updated with Changes: Keep yourself informed about any changes in the exam pattern or syllabus. This ensures that your preparation aligns with the current requirements. Stay Healthy Physically and Mentally: Physical and mental well-being are crucial during exam preparation. Ensure you are getting enough exercise, fresh air, and relaxation to keep stress levels in check. How to Prepare —Teach Others: Teaching a concept to someone else is a powerful way to reinforce your understanding. It helps solidify your knowledge and identify areas where you may need further clarification. Stay Consistent: Consistency is key in mathematics. Regular, small study sessions are often more effective than cramming. Stick to your schedule and avoid last-minute rushes. Reward Yourself: Celebrate small victories and milestones during your preparation. Rewarding yourself can boost motivation and make the studying process more enjoyable. Previous Years’ Papers: Solve previous years’ question papers to get a sense of the exam pattern and the types of questions asked. This also helps you practice time management. Visual Aids: Use visual aids such as charts, diagrams, and graphs to understand and remember complex concepts. Visual representation can make abstract ideas more tangible. How to Prepare—Stay Positive in Exam Hall: On the day of the exam, stay calm and positive. Take deep breaths if you feel anxious, and focus on the questions one at a time. Remember, everyone has their unique way of studying, so feel free to adapt these tips to suit your preferences and learning style. Good luck! How to Prepare Visit my YouTube channel: Click on the logo.

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ts inter multiplication of vectors 4 marks important questions

TS inter || Multiplication of Vectors 4m important questions

Education, TS Inter Maths, TS Inter Maths 1A / Nayini Satyanarayana Reddy

TS inter || Multiplication of Vectors 4m important questions Multiplication of Vectors Multiplication of vectors can take different forms depending on the context and the type of multiplication being used. Here are the main types: Scalar Multiplication: In scalar multiplication, a vector is multiplied by a scalar (a single number). Each component of the vector is multiplied by the scalar. For example, if you have a vector v = (x, y, z) and multiply it by a scalar k, you get kv = (kx, ky, kz). Here are some important questions related to the multiplication of Vectors that could be worth 4 marks each. Keep in mind that the specific marking scheme may vary based on the curriculum and exam format. These questions cover various aspects of the multiplication of vectors, including operations, properties, and applications: Maths IA Two-Mark Questions & Solutions Maths IB Two Marks Questions & Solutions Dot Product (Scalar Product): The dot product of two vectors produces a scalar. It is calculated by multiplying the corresponding components of the vectors and summing the results. For two vectors a and b, the dot product is denoted by a · b. The formula is: a · b = a₁b₁ + a₂b₂ + … + aₙbₙ. Geometrically, it represents the projection of one vector onto another. Cross Product (Vector Product): The cross product of two vectors results in another vector that is perpendicular to the plane containing the original vectors. It is denoted by a × b. The formula depends on the dimensionality of the vectors: For 3-dimensional vectors, the formula is a × b = (a₂b₃ – a₃b₂, a₃b₁ – a₁b₃, a₁b₂ – a₂b₁). The result is a vector perpendicular to both a and b, with a magnitude equal to the area of the parallelogram formed by a and b. These are the fundamental types of vector multiplication used in mathematics and physics. Each type has its properties and applications in various fields. Maths – IA Concept Maths – IB Concept Maths – IIA Concept Maths – IIB Concept Visit my YouTube channel: Click on the Logo

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TS Inter || Addition of Vectors 4 Marks Important Questions

Education, TS Inter Maths, TS Inter Maths 1A / Nayini Satyanarayana Reddy

TS Inter || Addition of Vectors 4 Marks Important Questions Addition of Vectors Vector addition is a fundamental operation in mathematics and physics, especially in the study of forces, velocities, and displacements. When you add vectors, you’re essentially combining their magnitudes and directions to find the resultant vector. Here’s how vector addition works: Magnitude Addition: To add the magnitudes of vectors, simply add their numerical values together. For example, if you have two vectors, A and B with magnitudes 3 and 4, respectively, their magnitudes add up to 7. Direction Addition: Vectors have both magnitude and direction. To add vectors, you must also consider their directions. You can represent vectors graphically using arrows, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction of the vector. Here are some important questions related to the addition of vectors that could be worth 4 marks each. Keep in mind that the specific marking scheme may vary based on the curriculum and exam format. These questions cover various aspects of the addition of vectors, including operations, properties, and applications: Addition of vectors 4 marks important questions Maths – IA Concept Maths – IB Concept Resultant Vector: The resultant vector is the sum of the individual vectors. To find the resultant vector, you can use methods like the parallelogram method, triangle method, or component method (using vector components). Parallelogram Method: This method involves constructing a parallelogram using the vectors to be added as adjacent sides. The diagonal drawn from the common point of the vectors represents the resultant vector. Triangle Method: If you have only two vectors, you can use the triangle method. Place the tail of the second vector at the head of the first vector, and draw a vector from the tail of the first vector to the head of the second vector. The resultant vector is the vector from the tail of the first vector to the head of the second vector. Component Method: You can break down vectors into their horizontal and vertical components. Then add the horizontal components separately and the vertical components separately. Finally, combine the horizontal and vertical components of the resultant vector to get the resultant vector. When adding vectors, it’s essential to maintain the correct signs (positive or negative) and directions. The resultant vector represents the net effect of all the vectors being added together. Maths – IIA Concept Maths – IIB Concept Visit my YouTube channel: Click on the logo.

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ts inter matrices 4 marks important questions 2024

TS Inter || Matrices 4 Marks Important Questions 2026

Education, TS Inter Maths, TS Inter Maths 1A / Nayini Satyanarayana Reddy

TS Inter || Matrices 4 Marks Important Questions 2026 Matrices Here are some important questions related to matrices that could be worth 4 marks each. Keep in mind that the specific marking scheme may vary based on the curriculum and exam format. These questions cover various aspects of matrices, including operations, properties, and applications: Visit my YouTube channel: Click on the Logo

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Rolles and Langranges Theorem vsaq’s (2 Marks)

Education, TS Inter Maths, TS Inter Maths1B / Nayini Satyanarayana Reddy

Rolle’s and Langranges Theorem Rolle’s and Langranges Theorem vsaq’s Rolle’s theorem If f: [a, b] ⟶ R be a function satisfying the following conditions f is continuous on [a, b] f is differentiable (a, b) f(a) = f(b) then there exists at least one cϵ (a, b) such that f’(c) = 0 Langrage’s theorem If f: [a, b] ⟶ R be a function satisfying the following conditions f is continuous on [a, b] f is differentiable (a, b) Then there exists at least one cϵ (a, b) such that f’(c) = Rolles and Langranges Theorem vsaq’s 1. State Rolle’s theorem Sol: If f: [a, b] ⟶ R be a function satisfying the following conditions f is continuous on [a, b] f is differentiable (a, b) f(a) = f(b) then there exists at least one cϵ (a, b) such that f’(c) = 0 2. State Langrage’s theorem Sol: If f: [a, b] ⟶ R be a function satisfying the following conditions f is continuous on [a, b] f is differentiable (a, b) Then there exists at least one cϵ (a, b) such that f’(c) = 3. If f(x) = (x – 1) (x – 2) (x – 3), prove that there is more than ‘c’ in (1, 3) such that f’ (c) = 0 Sol: Given function is f(x) = (x – 1) (x – 2) (x – 3) (i)f(x) is continuous on [1, 3] (ii) f(x) is differentiable on (1, 3) (iii) f(1) = (1 – 1) (1 – 2) (1 – 3) = 0 (– 1)( – 2) = 0 f(3) = (3 – 1) (3 – 2) (3 – 3) = (2) (1) (0) = 0 f(1) = f(3) f’(x) = (1 – 0) (x – 2) (x – 3) + (x – 1) (1 – 0) (x – 3) + (x – 1) (x – 2) (1 – 0) = (x – 2) (x – 3) + (x – 1) (x – 3) + (x – 1) (x – 2) = x2 – 3x – 2x + 6 + x2 – 3x – x + 3 + x2 – 2x – x + 2 = 3×2 – 12x + 11 f’ (c) = 3c2 – 12c + 11 by Rolle’s Theorem f’ (c ) = 0 3c2 – 12c + 11 = 0 4. Find all the values of ‘c’ in Rolle’s theorem for the function y = f(x) = x2 + 4 on [– 3, 3] Sol: Given function is f(x) = (x – 1) (x – 2) (x – 3) f(x) is continuous on [– 3, 3] f(x) is differentiable on (– 3, 3) f (– 3) = (– 3)2 + 4 = 9 + 4 = 13 f (3) = (3)2 + 4 = 9 + 4 = 13 f (– 3) = f(3) By Rolle’s theorem there exist c ∈ (– 3, 3) such that f’ (c) = 0 f’ (x) = 2x f’ (c) = 2c ⟹ 2c = 0 C = 0 ∈ (– 3, 3) 5. Find the value of ‘c’ from Rolle’s theorem for the function f(x) = x2 – 1 on [– 1, 1] Sol: Given function is f(x) = x2 – 1 f(x) is continuous on [– 1, 1] f(x) is differentiable on (– 1, 1) f (– 1) = (– 1)2 – 1 = 1 – 1 = 0 f (– 1) = (– 1)2 – 1 = 1 – 1 = 0 f (– 1) = f(1) By Rolle’s theorem there exist c ∈ (– 1, 1) such that f’ (c) = 0 f’ (x) = 2x f’ (c) = 2c ⟹ 2c = 0 c = 0 ∈ (– 1, 1) 6. It is given that Rolle’s theorem holds for the function f(x) = x3 + bx2 + ax on [1, 3] with c = 2 + . Find the values of a and b. Sol: Given function is f(x) = x3 + bx2 + ax on [1, 3] satisfying Rolle’s theorem ⟹ f(x) is continuous on [ 1, 3] f(x) is differentiable on (1, 3) f (1) = f (3) and there exists at least one c = 2 + ϵ (1, 3) such that f’ (c) = 0 now f(1) = f(3) 1 + b + a = 27 + 9b + 3a 2a + 8b + 26 = 0 a + 4b + 13 = 0 ———- (1) f’ (x) = 3 x2 + 2b x + a f’ (c) = 0 ⟹ 3 c2 + 2bc + a = 0 Since c = 2 + 3 + 2b ( ) + a = 0 3 + 4b ) + a = 0 3 + 4b ) + a = 0 + + 4b + + a = 0 13 + + 4b + + a = 0 ———- (2) Equation (2) – Equation (1) from (1) a + 4(– 6) + 13 = 0 a – 24 + 13 = 0 a = 11 ∴ a = 11, b = – 6 Rolles and Langranges Theorem vsaq’s 7. Verify Rolle’s theorem for the function f(x) = sin x – sin 2x on [0, π] Sol: Given function is f(x) = sin x – sin 2x f(x) is continuous on [0, π] f(x) is differentiable on (0, π) f(0) = sin (0) – sin 2(0) = 0 f( π) = sin ( π) – sin 2(π) = 0 f(0) = f( π) now f’ (x) = cos x – 2 cos 2x f ‘ (c) = 0 cos c – 2 cos 2c = 0 cos c – 2(2 cos2 c – 1) = 0 cos c – 4 cos2 c – 2 = 0 4 cos2 c – cos c + 2 = 0 c = ∈ (0, π) ∴ Rolle’s theorem is verified 8. Verify Rolle’s theorem for the function f(x) = (x2 – 1) (x – 2) on [ – 1, 2]

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Errors and Appraximations vsaqs questions and solutions

Errors and Approximations || V.S.A.Q’S||

Education, TS Inter Maths, TS Inter Maths1B / Nayini Satyanarayana Reddy

Errors and Approximations || V.S.A.Q’S|| Errors and approximations: These solutions were designed by the ‘Basics in Maths‘ team. These notes to do help intermediate First-year Maths students. Inter Maths – 1B two mark questions and solutions are very useful in IPE examinations. Errors and Approximations Question 1 Find dy and ∆y for the following functions for the values of x and ∆x which are shown against each of the functions (i) y = f(x) = x2 + x at x = 10 when ∆x = 0.1. Sol: Given y = f(x) = x2 + x at x = 10, ∆x = 0.1 ∆y = f (x + ∆x) – f (x) = f (10 + 0.1) – f (10) = f (10.1) – f (10) = (10. 1)2 + 10.1 – (102 + 10) = 102.01 + 10.1 – 100 – 10 = 112.11 – 110 = 2.11 dy = f’ (x) ∆x = (2x + 1) (0.1) = [2(10) + 1] (0.1) = 21 × 0.1 = 2.1 (ii) y = cos x at x = 600 with ∆x = 10 (10 = 0.0174 radians) Sol: Given y = cos x, x = 600 and ∆x = 10 ∆y = f (x + ∆x) – f (x) = cos (600 + 10) – cos 600 = cos (610) – 0.5 = 0.4848 – 0.5 = – 0.0152 dy = f’ (x) ∆x = – sin x (10) = – sin 600 × 0.0174 =– 0.8660 × 0.0174 = – 0.0150 (iii) y = x2 + 3x + 6, x = 10 with ∆x = 0.01 Sol: y = x2 + 3x + 6 ∆y = f (x + ∆x) – f (x) = f (10 + 0.01) – f (10) = f (10.01) – f (10) = (10.01)2 + 3 (10.01) + 6 – (102 + 3 (10) + 6) = 100. 2001 + 30.03 + 6 – 100 – 30 – 6 =130. 2301 – 130 = 0.2301 dy = f’ (x) ∆x = (2x + 3 + 0) (0.01) = (2× 10 + 3) (0.01) = 23 × 0.01 = 0.23 (iv) y = , x = 8 and ∆x =0.02 Sol: Question 2 The side of a square is increased from 3cm to 3.01cm find the approximate increase in the area of the square. Sol: Let x be the side of the square and the area be A Area of the square A = x2 x = 3 and ∆x = 0.01 ∆A = 2x × ∆x = 2(3) (0.01) = 6 × 0.01 = 0.06 Question 3 If an increase in the side of a square is 2% then find the approximate percentage of increase in its area. Sol: Let x be the side of the square and A be its area = 2 A = x2 ∆A = 2x × ∆x The approximate percentage error in area A = 2 × 2 =4 https://www.basicsinmaths.com/inter-mathematics-1a-and-1b-pdf-files/ Question 4 From the following. Find the approximations (i) Sol: Let f(x) = , where x = 1000 and ∆x =– 1 f’ (x) = Approximate value is f (x + ∆x) = f(x) + f’ (x) ∆x = 10 – 0. 0033 = 9.9967 (ii) Sol: (iii) Sol: (iv) Sin 620 Sol: Let f(x) = sin x, where x = 600 and ∆x =20 Approximate value is f (x + ∆x) = f(x) + f’ (x) ∆x = sin 600 + cos x (20) = sin 600+ cos 600 (0.0348) = 0.8660 + 0.5 × 0.0348 = 0.8660 + 0.0174 =0.8834 Question 5 The radius of a sphere is measured as 14cm. Later it was found that there is an error of 0.02cm in measuring the radius. Find the approximate error in the surface area of the sphere. Sol: Given r = 14 cm and ∆r =0.02cm Surface area of sphere =A = 4π r2 ∆A = 8π r ∆r = 8 ×3.14× 14 × 0.02 = 7.0336 Visit my Youtube Channel: Click on Below Logo

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TS inter 1st year Maths Blueprint

Education, TS Inter Maths / Nayini Satyanarayana Reddy

TS inter1st year Maths Blueprint TS Inter 1st year Maths Blueprint: : These blueprints were designed by the ‘Basics in Maths’ team. These to-dos help the TS intermediate first-year Maths students fall in love with mathematics and overcome their fear. These blueprints cover all the topics of the TS I.P.E first-year maths syllabus and help in I.P.E exams. Maths IA Two-Mark Questions & Solutions Maths IB Two Marks Questions & Solutions Maths – IA Concept Maths – IB Concept TS inter 1st year Maths Blueprin Maths – IIA Concept Maths – IIB Concept YouTube

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TS Inter second year Maths 2B Concept

Education, TS Inter Maths, TS Inter Maths 2B / Nayini Satyanarayana Reddy

TS Inter second year TS Inter second year: This note is designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate second-year Maths students fall in love with mathematics and overcome the fear. These notes cover all the topics covered in the TS I.P.E second year maths 2B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations. TS Inter second year 1. CIRCLES Circle: In a plane, the set of points that are at a constant distance from a fixed point is called a circle. ∗ The fixed point is called the centre (C) of the circle and the constant distance is called the radius(r) of the circle Unit circle: If the radius of the circle is 1 unit, then that circle is called the unit circle. Point Circle: A circle is said to be a point circle if its radius is zero. A point circle contains only one point in the centre of the circle. • ∗ The equation of the circle with centre (h, k) and radius r is (x – h)2 + (y – k)2 = r2 ∗ The equation of the circle with centre origin and radius r is x2 + y2 = r2 ⇒ x2 + y2 = r2 is called standard form of the circle. The general equation of the second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, where a, b, f, g, h and c are real numbers, represent a circle iff (i) a = b ≠ 0 (ii) h = 0 and (iii) g2 + f2 + c ≥ 0 ∗ The general equation of the circle is x2 + y2 + 2gx + 2fy + c = 0 It’s centre c = (– g, – f) and radius ∗ The equation of the circle passing through origin is x2 + y2 + 2gx + 2fy = 0. ∗ The equation of the circle whose centre on the x-axis is x2 + y2 + 2gx + c = 0. ∗ The equation of the circle having centre on y-axis is x2 + y2 + 2fy + c = 0. ∗ The circles which have the same centre are called concentric circles. ∗ The equation of the circle concentric with the circle x2 + y2 + 2gx + 2fy + c = 0 is x2 + y2 + 2gx + 2fy + k = 0. ∗ The length of the intercept made by a circle x2 + y2 + 2gx + 2fy + c = 0 on x -axis is if g2 – c > 0 y -axis is if f2 – c > 0 Note: – (a) if g2 – c = 0, then A1 A2 = 0 ⇒ the circle touches the x- axis at only one point. (b) if f2 – c = 0, then B1 B2 = 0 ⇒ the circle touches the y- axis at only one point. (c) if g2 – c < 0, then the circle does not meet the x- axis. (d) if f2 – c < 0, then the circle does not meet the y- axis. ∗ The equation of the circle having the line segment joining A (x1, y1) and B (x2, y2) as a diameter is (x – x1) (x – x2) + (y – y1) (y – y2) = 0. ∗ Let A, B be any two points on a circle then, The line is called the secant line of the circle. The line segment is called the chard of the circle. AB is called the length of the chord. ∗ A chord passing through the centre is called the diameter of the circle. ∗ The angle subtended by a chord on the circumference of at any point is equal. The perpendicular bisector of a chord of a circle is asses through the centre of the circle. ∗ The angle in a semicircle is 900. ∗ The equation of the circle passing through three non-collinear points A (x1, y1), B (x2, y2), C (x3, y3) is Where ci = − (x2 + y2) and i = 1,2,3 ∗ centre of the circle is Parametric form: If P (x, y) is a point on the circle with centre (h, k) and radius r, then X = h + r cosθ, y = k + r sinθ 0 ≤ θ ≤ 2π. ⇒ A point n the circle x2 + y2 = r2 is taken as (r cosθ, r sinθ) and simply denoted by θ. Note: If the centre of the circle is the origin, then the parametric equations are x = r cosθ, y = r, 0 ≤ θ ≤ 2π. The point (h + rcosθ1, k + r sin θ1) is referred to as the point θ1 on the circle having the centre (h, k) and radius r. Notations: S = x2 + y2 + 2gx + 2fy + c S1 = xx1 + yy1 + g(x +x1) + f (y +y1) + c S11 = x12 + y12 + 2gx1 +2fy1 + c S12 = x1x2 + y1y2 + g(x1 + x2 ) + f (y1 + y2) + c Position of a point with respect to the circle: A circle divides the plane into three parts. 1. The interior of the circle 2. The circumference which is the circular curve. 3. The exterior of the circle. Power of point: Les S = 0 be a circle with radius ‘r’ and centre ‘C’ and P (x1, y1) be a point on the circle, then CP – r2 is called the power of point ‘P’ concerning S = 0. The power of point P (x1, y1) w.r.t. S = 0 is S11. •Let S = 0 be a circle in a plane and P

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