Question

The sum of diameters of the circles that touch (i) the parabola 75 x 2 = 64(5 y – 3) at the point ($\frac{8}{5},\frac{6}{5}$) and (ii) the y -axis is equal to _______.

Answer 1

Equation of tangent to the parabola at
P($\frac{8}{5},\frac{6}{5}$)
75x⋅85=160(y+$\frac{6}{5}$)−192
⇒ 120 x = 160 y
⇒ 3 x = 4 y
The equation of the circle touching the given parabola at P can be taken as
(x−$\frac{8}{5}$)2+(y−$\frac{6}{5}$)2+λ(3x−4y)=0
If this circle touches the y -axis then
$\frac{64}{25}$+(y−65)2+λ(−4y)=0
⇒y2−2y(2λ+65)+4=0
⇒ D = 0
⇒($\frac{2λ+6}{6}$)2=4
⇒λ=$\frac{2}{5}$ or −$\frac{8}{5}$
Radius = 1 or 4
Sum of diameter = 10