The locus of the point of intersection of the tangents at the extremities of the chords of the ellipse `x^2+2y^2=6` which touch the ellipse `x^2+4y^2=4,` is `x^2+y^2=4` (b) `x^2+y^2=6` `x^2+y^2=9` (d) None of these

Options

A. `x^(2)+y^(2)=4`
B. `x^(2)+y^(2)=6`
C. `x^(2) +y^(2) =9`
D. None of these

Answer

C

Explanation

The given equation of second ellipse can be rewritten as
` (x^2)/4 + y^(2)/1 =1`
Equation of tangent to this ellispe is
` x/2 cos theta + y sin theta =1`
Equation of the first ellipse can be rewritten as
` x^(2)/6 + y^(2)/3=1`
Let Eq. (i) Meets the first ellipse at P and Q and the tangents at P and Q to the second ellispe intersected at (h,k) then Eq.(i) is the chord of contact of (h,k) with respect to the ellispe (ii) and thus, its equation is
`(hx)/6, + (ky)/3 =1 `
Since ,Eqs (i) and (iii) represent the same line
`(h//6)/(cos theta)/2 = (k//3)/(sint theta) =1`
` Rightarrow h = 3 cos theta`
` and k = 3 sin theta `
Hence , locus is ` x^(2) , + y^(2) = 9`