NTA Abhyas JEE Main2020MathematicsHyperbolaPractice

The locus of the point of intersection of two tangents of the hyperbola x 2 2 - y 2 4 = 1 , if the product of their slopes is 1 , is

Options

Ay 2 - 4 = x 2 + 2
By 2 + 4 = x 2 - 2
Cy 2 + 2 = x 2 - 4
Dy 2 - 2 = x 2 + 4

Correct answer

B. y 2 + 4 = x 2 - 2

Step-by-step solution

Equation of any tangent of the hyperbola with slope m is y = m x ± 2 m 2 - 4 If it passes through x 1 , y 1 then y 1 - m x 1 2 = 2 m 2 - 4 ⇒ x 1 2 - 2 m 2 - 2 x 1 y 1 m + y 1 2 + 4 = 0 If n = m 1 , m 2 then as given m 1 m 2 = 1 ⇒ y 1 2 + 4 x 1 2 - 2 = 1 Hence, the required locus is y 2 + 4 = x 2 - 2

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