Question

Let a line L₁ be tangent to the hyperbola
\(\frac{x²}{16} - \frac{y²}{4} = 1\)
and let L₂ be the line passing through the origin and perpendicular to L₁. If the locus of the point of intersection of L₁ and L₂ is
\(( x² + y²)² = αx² + βy²,\)
then α + β is equal to___.

Answer 1

Correct Answer: 12

The correct answer is 12
Equation of L₁ is
\(\frac{xsecθ}{4} - \frac{ytanθ}{2} = 1 ...... (i)\)
Equation of line L₂ is
\(\frac{x tanθ}{2} + \frac{y secθ}{4} = 0 ....... (ii)\)
∵ Required point of intersection of L₁ and L₂ is (x₁, y₁) 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 (x₁, y₁) is
\(( x² + y² )² = 16x² - 4y²\)
∴ α = 16, β = -4
Therefore, α + β = 12