Question

$${\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 1

$${\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{.}} \\\$$