Equations of two diameters of a circle are 2 x - 3 y = 5 and 3 x - 4 y = 7 . The line joining the points - 22 7 , - 4 and - 1 7 , 3 intersects the circle at only one point P ( α , β ) . Then 17 β - α is equal to

Step-by-step solution

Given diameters of circle are 2 x - 3 y = 5 and 3 x - 4 y = 7 . We know that, the intersection point of diameters is centre. ⇒ x = 5 + 3 y 2 = 7 + 4 y 3 ⇒ 15 + 9 y = 14 + 8 y ⇒ y = - 1 ⇒ x = 5 - 3 2 = 1 So, centre of circle is 1 , - 1 Equation of A B is given by, y + 4 = 3 + 4 - 1 7 + 22 7 x + 22 7 ⇒ y + 4 = 7 3 7 x + 22 7 ⇒ 3 y + 12 = 7 x + 22 ⇒ 7 x – 3 y + 10 = 0 . . . i Slope of A B , m 1 = 7 3 So, slope of C P , m 2 = - 1 m 1 = - 3 7 Equation of C P is given by, y + 1 = - 3 7 x - 1 ⇒ 7 y + 7 = - 3 x + 3 ⇒ 3 x +