The locus of the point of intersection of perpendicular tangents to `x^(2)/a^(2) + y^(2)/b^(2) = 1` and `(x^(2))/(a^(2) + lambda) + (y^(2))/(b^(2) + lambda) = 1`, is

Options

A. `x^(2) + y^(2) = a^(2) + lambda`
B. `x^(2) + y^(2) = b^(2) + lambda`
C. `x^(2) + y^(2) = a^(2) + b^(2) + lambda`
D. `x^(2) + y^(2) = a^(2) + b^(2)`

Answer

C

Explanation

Equation of any tangent to `x^(2)/a^(2) + y^(2) /b^(2) = 1` is
`y = mx + sqrt(a^(2)m^(2) + b^(2))" "…(i)`
The equation of tangent to `(x^(2))/(a^(2) + lambda) + (y^(2))/(b^(2) + lambda) = 1` perpendicular to (i) is
`y - 1/m x + sqrt((a^(2) + lambda) 1/m^(2) + (b^(2) + lambda))`
`rArr my = -x + sqrt((a^(2) + lambda) + (b^(2) + lambda)m^(2))" "...(ii)`
Let P (h, k) be the point of intersection of (i) and (ii). Then,
`(k - mh)^(2) = a^(2)m^(2) + b^(2) and (mk + h)^(2) = (a^(2) + lambda) + (b^(2) + lambda)m^(2)`
`rArr (k - mh)^(2) + (mk + h)^(2) = (a^(2) + b^(2) + lambda)(m^(2) + 1)`
`rArr h^(2) + k^(2) = a^(2) + b^(2) + lambda`
Hence, the locus of (h, k) is `x^(2) + y^(2) = a^(2) + b^(2) + lambda.`