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Conic Sections
Parabola
Medium

Question

Let L 1 be a tangent to the parabola y 2 = 4(x + 1) and L 2 be a tangent to the parabola y 2 = 8(x + 2) such that L 1 and L 2 intersect at right angles. Then L 1 and L 2 meet on the straight line :

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Solution

Key Concepts: Equation of Tangent to a Parabola and Conditions for Perpendicular Lines

The core of this problem involves two main mathematical concepts:

  1. Equation of Tangent to a Parabola: For a standard parabola in the form Y2=4aXY^2 = 4aX, the equation of a tangent line with a slope mm is given by: Y=mX+amY = mX + \frac{a}{m} This formula is incredibly versatile. When a parabola is shifted, for example, from y2=4axy^2 = 4ax to y2=4a(x−h)y^2 = 4a(x-h), we simply replace the standard coordinate XX with the shifted coordinate (x−h)(x-h) in the general tangent equation. If the YY coordinate is also shifted, say to (y−k)2=4aX(y-k)^2 = 4aX, then $
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