Hyperbola Equation

Hyperbola Equation Let S be a fixed point and L, a fixed straight line on a plane. If a point P moves on this plane in such a way that its distance from the fixed point S always bears a constant ratio to its perpendicular distance from the fixed line L and if this ratio is greater than unity then the locus traced out by P is called a hyperbola. Equation of a hyperbola whose center is at origin (0, 0) is x^2 / a^2 y^2 / b^2 = 1 (1) Question 1: - Find the lengths of axes of the parabola 9 x^2 25 y ^2 = 225. Solution: - 9 x^2 25 y ^2 = 225. x^2 / 25 y^2 / 9 = 1 (2) Comparing equation (2) with the standard form of hyperbola (1) we get, A^2 = 25 or, a = 5 and b^2 = 9 or, b = 3 Therefore, the length of the transverse axis of the hyperbola (2) is 2 a = 2 * 5= 10 And the length of the conjugate axis = 2 b = 2 * 3 = 6. Question 2: - If length of the transverse and conjugate axes of a hyperbola is 8 and 12 respectively, then find the equation of the hyperbola. Solution: - According to the problem, 2 a = 8, therefor a = 4 And 2 b = 12, therefor b = 6. Substituting these values in equation (1) we get, x^2 / 4^2 y^2 / 6^2 = 1 x^2/16 y^2/ 36 = 1 9x^2 4y^2 =144