The is a point P on the hyperbola `(x^(2))/(16)-(y^(2))/(6)=1` such that its distance from the right directrix is the average of its distance from the two foci. Then the x-coordinate of P is

Options

A. the locus of excenter of the circle described opposite to `angleP` for `DeltaPSS'` (S, S" are foci) is tangent at vertex
B. the locus of the excenter of the circle described opposite to `angleS'` is a hyperbola
C. the locus of the excenter of the circle described opposite to `angleP` for `DeltaRSS'` (S, S' are foci) is a hyperbola
D. the locus of the excenter of the circle described opposite to `angleS'` is tangent at vertex.

Answer

A, B

Explanation

(1), (2)
Let (h, k) be the excenter. Then,
`h=(ae(ae sec theta+a)-ae(ae sectheta-a)-2ae(a sec theta))/(2ae(sec theta-1))=-a` ltBrgt `"or "x=-a("for "S'Pgt SP)` ltbRgt Similarly, x = a (for S'P lt SP).
Therefore, the locus is `x^(2)=a^(2)`

Again, let (h, k) be the excenter opposite `angleS'`. Then ,
`h=(2a^(2)e sec theta+a^(2)e^(2)sectheta+a^(2)e^(2) sec theta-a^(2)e)/(2a+2ae)`
`aesec theta`
`"and "k=(2aeb tan theta)/(2a+2ae)`
Therefore, the locus is a hyperbola.