If `alpha-beta=` constant, then the locus of the point of intersection of tangents at `P(acosalpha,bsinalpha)` and `Q(acosbeta,bsinbeta)` to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` is a circle (b) a straight line an ellipse (d) a parabola

Options

A. a circle
B. a straight line
C. an ellipse
D. a parabola

Answer

C

Explanation

Let R (h, k) be point ofi intersection of tangents at P and Q. Then,
`h = (a cos ((alpha + beta)/(2)))/(cos ((alpha - beta)/(2))) and k = (b sin ((alpha + beta)/(2)))/(cos ((alpha - beta)/(2)))`
`rArr h^(2)/a^(2) + k^(2)/b^(2) = (1)/(cos^(2)((alpha - beta)/(2)))`
Hence, the locus of R (h, k) is
`x^(2)/a^(2) + y^(2)/b^(2) = (1)/(cos^(2)((alpha - beta)/(2)))" "[because alpha beta = ` constant]
Clearly, it represents an ellipse.