JEE Main 2023 (Online) 24th January Morning ShiftQuestion 1
Let a tangent to the curve $y^{2}=24x$ meet the curve $xy=2$ at the points $A$ and $B$. Then the mid-points of such line segments $AB$ lie on a parabola with the
Parabola
Solution
Concept. Write the chord of $xy=2$ with midpoint $(h,k)$ and match it with the tangent of the parabola. Formulas used. (1) Tangent to $y^2=24x$: $y=mx+\dfrac6m$.(2) Chord of $xy=2$ with midpoint $(h,k)$ ($T=S_1$): $kx+hy=2hk$.(3) $y^2=-4Ax$: directrix $x=A$, latus rectum $4A$.Step 1 - Use (2). $y=-\dfrac khx+2k$: slope $m=-\dfrac kh$, intercept $2k$. Step 2 - Use (1). Intercept $=\dfrac6m$: $2k=6\cdot\Big(-\dfrac hk\Big)\Rightarrow k^2=-3h$. Step 3 - Locus. $y^2=-3x$: $4A=3$, $A=\frac34$. Step 4 - Use (3). Directrix $x=\frac34$, i.e. $4x=3$. Latus rectum $3$, so (C) and (D) are false. Answer: A