Coordinate Geometry
Pair of lines cutting ellipse; locus of intersection of tangents
MMTS_Full_Test_14
Grade 12
Question:
If pair of variable straight lines $x^2+4y^2+\alpha xy=0$ ($\alpha$ real) cuts the ellipse $x^2+4y^2=4$ at $A$ and $B$, then the locus of point of intersection of tangents at $A$ and $B$ of ellipse is
(A) $x^2-4y^2=0$
(B) $4x^2-y^2=0$
(C) $x^2-xy+4y^2$
(D) $x^2-xy-4y^2=0$
Step-by-Step Solution
Key Concept: Chord of contact from $(h,k)$ to $x^2+4y^2=4$ is $hx+4ky=4$. This chord lies on $x^2+4y^2+\alpha xy=0$: the two lines pass through origin and their equation is $x^2+4y^2+\alpha xy=0$. The chord $AB$ is the polar. Combined: $h^2-4k^2=0$ (homogenizing).
Locus: $x^2-4y^2=0$.
Correct Answer: (A) $x^2-4y^2=0$