Question

If a pair of variable straight lines `x^2 + 4y^2+alpha xy =0` (where `alpha` is a real parameter) cut the ellipse `x^2+4y^2= 4` at two points A and B, then the locus of the point of intersection of tangents at A and B is
A. `x-2y=0`
B. `2x-y=0`
C. `x+2y=0`
D. `2x+y=0`

Answer 1

Let the point of intersection of tangents A and B be P(h,k).Then the equation of AB is
`(xh)/(4)+(yk)/(1)=1" "(1)`

Homogennizing ithe equation of ellipse using (1), we get
` (x^(2))/(4)+(y^(2))/(1)=((xh)/(4)+(yk)/(1))^(2)`
or `x^(2)((h^(2)-h)/(16))+y^(2)(k^(2)-1)+(2hk)/(4)xy=0" "(2)`
The given equation of OA and OB is
`x^(2)+4y^(2)+alphaxy=0" "(3)`
Sinc (2) and (3) repreetn the same line,
`(h^(2)-4)/(16)+(k^(2)-1)/(4)=(hk)/(2a)`
`or h^(2)-4=4(k^(2)-1)`
or `h^(2)-4k^(2)=0`
Therefore, the locus is `(x-2y)(x+2y)=0`