Hyperbola
Locus of Midpoints / Parabola
nta_pyq_2023_jan
Grade 11
Question:
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
directrix $4x = 3$
directrix $4x = -3$
Length of latus rectum $\dfrac{3}{2}$
Length of latus rectum $2$
Step-by-Step Solution
Key Concept: Tangent to $y^2=24x$ is $ty=x+6t^2$; intersect with $xy=2$, find midpoint locus.
Tangent: $ty=x+6t^2$. Intersecting $xy=2$ and finding midpoint locus gives $y^2=-3x$. Directrix of this parabola: $4x=3$.
Correct Answer: 1