The sum of diameters of the circles that touch (i) the parabola 75x2=64(5y−3) at the point (58,56) and (ii) the y-axis, is equal to \_\_\_\_\_\_\_\_.
Answer: 10
Solutions · 1
Official solution
The tangent to the parabola at (58,56) has direction such that the circle's line of centres is normal there, giving normal direction 3x−4y=0. A circle touching the parabola at that point can be written
\[\left(x-\frac85\right)^2+\left(y-\frac65\right)^2+\lambda(3x-4y)=0.\]
For tangency to the y-axis, set x=0 and require the resulting quadratic in y to have a repeated root:
\[y^2-2y\left(2\lambda+\frac65\right)+4=0,\qquad \left(2\lambda+\frac65\right)^2=4\ \Longrightarrow\ \lambda=\frac25\ \text{or}\ -\frac85.\]
These give radii 1 and 4 respectively, so the sum of diameters is 2(1)+2(4)=10.