Prove that the focus of id-points of the portion of the tamgents to the ellipse `x^(2)/a^(2)+y^(2)/b^(2)=1` intercepted between the axes is a `a^(2)y^(2)+b^(2)x^(2)=4x^(2)y^(2)`.

Options

A. `(x^(2))/(a^(2))+(y^(2))/(b^(2))=4`
B. `(a^(2))/(x^(2))+(b^(2))/(y^(2))=4`
C. `(x^(2))/(a^(2))-(v^(2))/(a^(2))=4`
D. none of these

Answer

B

Explanation

the equation of the tangent at any point `phi ` of the ellipse is
`(x)/(a) cos phi +(y)/(b) sin phi =1`
It meets the coordinate axes at the points A `(a sec phi ,0) and B(0,b cosec phi)`
Let (h,k) be the mid point of AB then
` h=(a secphi)/(2),k=( b cos phi)/(2)`
`implies cos phi =(a)/(2h) and sin phi =(b)/(2k)implies (a^(2))/(4h^(2))+(b^(2))/(4k^(2))=1`
hence , the locus of `(a^(2))/(x^(2))+(b^(2))/(y^(2))=4.`