Question

If a parabola touch three given straight lines, prove that each of the lines joining the points of contact passes through a fixed point.

Answer 1

If two of the tangents are the axes then equation of parabola is

$\sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=1$
If the third tangent is $\frac{x}{f}+\frac{y}{g}=1$

The condition for tangency is

$\frac{f}{a}+\frac{g}{b}=1$

So the line $\frac{x}{a}+\frac{y}{b}=1$ always passes through $(f,g)$

Hence proved.