Let a tangent to the curve meet the curve at the points A and B. Then the mid points of such line segments AB lie on a parabola with the
- A.directrix
- B.length of latus rectum
- C.directrix
- D.length of latus rectum
Show answer and solution
Correct option: C
Solution
The tangent to y² = 24x at parameter t is: ty = x + 6t².
Let midpoint of AB on xy = 2 be (h, k). For a chord of xy = 2 with midpoint (h, k), the chord equation (using T = S₁ for rectangular hyperbola) is:
x·k + y·h = 2hk, i.e., kx + hy = 2hk.
Since this chord is the tangent ty = x + 6t², rewrite it as: -x + ty = 6t².
Comparing -x + ty = 6t² with kx + hy = 2hk:
(-1)/k = t/h = 6t²/(2hk)
From (-1)/k = t/h: t = -h/k.
From t/h = 6t²/(2hk): 1/h = 6t/(2hk) = 3t/(hk), so k = 3t.
Since t = -h/k = -h/(3t), we get 3t² = -h, so h = -3t².
And k = 3t.
So k² = 9t² = -3h → k² = -3h.
The locus is y² = -3x, a parabola opening left. Here 4a = 3, so a = 3/4.
The directrix is at x = a = 3/4, i.e., 4x = 3.
The length of latus rectum is 3 (not 3/2 or 2), confirming option 3 is correct: directrix 4x =
Let midpoint of AB on xy = 2 be (h, k). For a chord of xy = 2 with midpoint (h, k), the chord equation (using T = S₁ for rectangular hyperbola) is:
x·k + y·h = 2hk, i.e., kx + hy = 2hk.
Since this chord is the tangent ty = x + 6t², rewrite it as: -x + ty = 6t².
Comparing -x + ty = 6t² with kx + hy = 2hk:
(-1)/k = t/h = 6t²/(2hk)
From (-1)/k = t/h: t = -h/k.
From t/h = 6t²/(2hk): 1/h = 6t/(2hk) = 3t/(hk), so k = 3t.
Since t = -h/k = -h/(3t), we get 3t² = -h, so h = -3t².
And k = 3t.
So k² = 9t² = -3h → k² = -3h.
The locus is y² = -3x, a parabola opening left. Here 4a = 3, so a = 3/4.
The directrix is at x = a = 3/4, i.e., 4x = 3.
The length of latus rectum is 3 (not 3/2 or 2), confirming option 3 is correct: directrix 4x =
[JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)]