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

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

$ \sqrt {\dfrac{x}{a}} + \sqrt {\dfrac{y}{b} = 1} $

If the third tangent is $\dfrac{x}{f} + \dfrac{y}{g} = 1$

Then, the condition for the tangency is

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

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

Note: In this type of question always start with two static straight lines then introduce a third line with condition.

Last updated date: 04th Jun 2023

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