TS Inter second year Maths 2B Concept
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