Question

Let a line L 1 be tangent to the hyperbola
$\frac{x²}{16} - \frac{y²}{4} = 1$
and let L 2 be the line passing through the origin and perpendicular to L 1. If the locus of the point of intersection of L 1 and L 2 is
$( x² + y²)² = αx² + βy²,$
then α + β is equal to___.

Answer 1

The correct answer is 12
Equation of L 1 is
$\frac{xsecθ}{4} - \frac{ytanθ}{2} = 1 ...... (i)$
Equation of line L 2 is
$\frac{x tanθ}{2} + \frac{y secθ}{4} = 0 ....... (ii)$
∵ Required point of intersection of L 1 and L 2 is (x 1, y 1) then
$\frac{x_1secθ}{4} - \frac{y_1tanθ}{2} - 1 = 0 ...... (iii)$
and
$\frac{y_1secθ}{4} + \frac{x_1tanθ}{2} = 0 ....... (iv)$
From equations (iii) and (iv), we get
$\sec\theta = \frac{4x_1}{x_1^2 + y_1^2}$ and $\tan\theta = \frac{-2y_1}{x_1^2 + y_1^2}$
∴ Required locus of (x 1, y 1) is
$( x² + y² )² = 16x² - 4y²$
∴ α = 16, β = -4
Therefore, α + β = 12