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

Correct Answer: 10

\(x^2\)=\(\frac{64.5}{75}\)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