1.

If p is a point on a hyperbola, then

Answer»

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

Solution :(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.


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