JEE Main 2022 · 26 Jun · Shift 2 · Mathematics Q29

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MathematicsInteger+4 / -1Question 29 of 90

Let a line $L_1$ be tangent to the hyperbola $\dfrac{x^2}{16}-\dfrac{y^2}4=1$ 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 $\left(x^2+y^2\right)^2=\alpha x^2+\beta y^2$ , then $\alpha+\beta$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 12

Foot of the perpendicular $(h,k)$ from O to the tangent: the tangent is $hx+ky=h^2+k^2$ , slope $m=-\frac hk$ , intercept $c=\frac{h^2+k^2}k$ .

Tangency $c^2=16m^2-4$ : $(h^2+k^2)^2=16h^2-4k^2$ .

So $\alpha=16$ , $\beta=-4$ , $\alpha+\beta=12$ .

Answer: 12

Topics:Tangents and normals to conics Locus

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JEE Main 2022 · 26 Jun · Shift 2, Mathematics question 29