The locus of the mid points of the chords of the hyperbola x² - y² = 4, which touch the parabola y² = 8x, is :

Solution

Maths sir · 3 steps

Write the chord with midpoint (h, k) using T = S₁, then make that line a tangent to the parabola.

1. Chord from midpoint. For x² - y² = 4, the chord with midpoint (h, k) is T = S₁, that is hx - ky = h² - k². In slope form this is y = h/kx + (k² - h²)/k.
2. Tangency condition. For y² = 8x we have a = 2. The line y = mx + c touches it when c = a/m = 2/m. Here m = h/k and c = (k² - h²)/k.

   (k² - h²)/k = 2k/h ⇒ hk² - h³ = 2k² ⇒ k²(h - 2) = h³

3. So. Replace (h, k) by (x, y). The locus is y²(x - 2) = x³, which is option (c).

The idea to keep. For the locus of midpoints of chords, write the chord as T = S₁ first. Then apply whatever extra condition the question gives, here the tangency condition c = a/m for y² = 4ax.

Answer from the NTA final answer key. Solution by PrepWiser.