Question

The locus of the point of intersection of tangents to the hyperbola $4x^{2} - 9y^{2} = 36$ which meet at a constant angle $\pi/4$, is

• A. $(x^{2} + y^{2} - 5)^{2} = 4(9y^{2} - 4x^{2} + 36)$
• B. $(x^{2} + y^{2} - 5) = 4(9y^{2} - 4x^{2} + 36)$
• C. $4(x^{2} + y^{2} - 5)^{2} = (9y^{2} - 4x^{2} + 36)$
• D. None of these

Answer 1

Answer: (A)

$(x^{2} + y^{2} - 5)^{2} = 4(9y^{2} - 4x^{2} + 36)$

Answer 2

Let the point of intersection of tangents be $P(x_{1},y_{1})$. Then the equation of pair of tangents from $P(x_{1},y_{1})$ to the given hyperbola is

$(4x^{2} - 9y^{2} - 36)(4x_{1}^{2} - 9y_{1}^{2} - 36) = \lbrack 4x_{1}x - 9y_{1}y - 36\rbrack^{2}$......(i)

From $SS_{1} = T^{2}$ or $x^{2}(y_{1}^{2} + 4) + 2x_{1}y_{1}xy + y^{2}(x_{1}^{2} - 9) + ..... = 0$.....(ii)

Since angle between the tangents is $\pi/4$.

∴ $\tan(\pi/4) = \frac{2\sqrt{\lbrack x_{1}^{2}y_{1}^{2} - (y_{1}^{2} + 4)(x_{1}^{2} - 9)\rbrack}}{y_{1}^{2} + 4 + x_{1}^{2} - 9}$.

Hence locus of $P(x_{1},y_{1})$ is $(x^{2} + y^{2} - 5)^{2} = 4(9y^{2} - 4x^{2} + 36)$.