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Q.
The line $3x + 2y + 1 = 0$ meets the hyperbola $4x^2-y^2 = 4a^2$ in the points P and Q. The coordinates of the point of intersection of the tangents at P and Q are
Conic Sections
Solution:
Let the required point be $(x_1, y_1)$. The given line
$3x + 2y + 1 = 0 \quad ...(1)$
is chord of contact of the point so it must be same as the line
$T = 0,\, i.e.\, 4xx_1 - yy_1 = 4a^2 \quad ...(2)$
Comparing the coefficients of (1) and (2), we get
$\frac{4x_{1}}{3} = \frac{-y_{1}}{2} = \frac{4a^{2}}{-1} \Rightarrow x_{1} = -3a^{2}, y_{1} = 8a^{2}$