# \[

{\text{. If a parabola touch three given straight lines, prove that each of the lines joining the points }} \\

{\text{of contact passes through a fixed point}}{\text{.}} \\

\]

Answer

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369.6k+ views

\[

{\text{If two of the tangents are the axis then equation of parabola is}} \\

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

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

{\text{Then, the condition for the tangency is}} \\

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

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

{\text{Note: In this type of question always start with two static straight lines}} \\

{\text{ then introduce third line with condition}}{\text{.}} \\

\]

{\text{If two of the tangents are the axis then equation of parabola is}} \\

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

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

{\text{Then, the condition for the tangency is}} \\

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

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

{\text{Note: In this type of question always start with two static straight lines}} \\

{\text{ then introduce third line with condition}}{\text{.}} \\

\]

Last updated date: 30th Sep 2023

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