Question

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

Answer 1

$x^2$=$\frac{64.5}{75}$y−$\frac{3}{5}$
equation of the tangent at $\frac{8}{5}$,$\frac{6}{5}$
x⋅$\frac{8}{5}$=$\frac{64}{15}$ y+$\frac{6}{\frac{5}{2}}$−$\frac{3}{5}$
3x – 4y = 0
equation of a family of circles is
x−$\frac{8^2}{5}$+y−$\frac{6^2}{5}$+λ(3x−4y)=0
It touches the y-axis so $f^2$ = c
$x^2$+$y^2$+x3λ−$\frac{16}{5}$+y−4λ−$\frac{12}{5}$+4=0
$\frac{4λ+\frac{12^2}{5}}{4}$=4
λ=$\frac{2}{5}$or λ=−$\frac{8}{5}$
λ=$\frac{2}{5}$,r=1
λ=−$\frac{8}{5}$,r=4
d1+d2=10