JEE Main 2022 · 26 Jun · Shift 1 · Mathematics Q23

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MathematicsInteger+4 / -1Question 23 of 90

Let the common tangents to the curves $4\left(x^2+y^2\right)=9$ and $y^2=4x$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\dfrac l{e^2}$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 4

Common tangents. $y=mx+\frac1m$ touches $x^2+y^2=\frac94$ when $\dfrac{1}{|m|\sqrt{1+m^2}}=\dfrac32$ , i.e. $m^2=\frac13$ .

Point Q. The tangents $y=\pm\frac x{\sqrt3}\pm\sqrt3$ meet at $(-3,0)$ , so $OQ=3$ .

Ellipse $b=3$ , $a=6$ : $e^2=\frac34$ , $l=\frac{2b^2}a=3$ , so $\dfrac l{e^2}=4$ .

Answer: 4

Topics:Ellipse Tangents and normals to conics Tangents and normals

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JEE Main 2022 · 26 Jun · Shift 1, Mathematics question 23