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JEE Main 2023
Conic Sections
Ellipse
Medium

Question

Let a line L pass through the point of intersection of the lines bx+10y−8=0b x+10 y-8=0 and 2x−3y=0, b∈R−{43}2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right\}. If the line L\mathrm{L} also passes through the point (1,1)(1,1) and touches the circle 17(x2+y2)=1617\left(x^{2}+y^{2}\right)=16, then the eccentricity of the ellipse x25+y2 b2=1\frac{x^{2}}{5}+\frac{y^{2}}{\mathrm{~b}^{2}}=1 is :

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Solution

Key Concepts and Formulas

This problem integrates several fundamental concepts from coordinate geometry. A clear understanding of these is crucial for a systematic approach:

  1. Family of Lines: The equation of any line passing through the point of intersection of two given lines L1:A1x+B1y+C1=0L_1: A_1x + B_1y + C_1 = 0 and L2:A2x+B2y+C2=0L_2: A_2x + B_2y + C_2 = 0 is given by L1+λL2=0L_1 + \lambda L_2 = 0, or (A1x+B1y+C1)+λ(A2x+B2y+C2)=0(A_1x + B_1y + C_1) + \lambda(A_2x + B_2y + C_2) = 0, where λ\lambda is an arbitrary real constant
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