Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
If P is a point on a hyperbola, then
Locus of excentre of the circle described opposite to $\angle P$ for $\triangle PSS'$ (S, S' are foci), is tangent at vertex
Locus of excentre of the circle described opposite to $\angle S'$ is hyperbola
Locus of excentre of the circle described opposite to $\angle P$ for $\triangle PSS'$ (S, S' are foci), is hyperbola
Locus of excentre of the circle described opposite to $\angle S'$, is tangent at vertex
Step-by-Step Solution
Key Concept: For a hyperbola with foci S and S', when P is on the hyperbola, the locus of excenters of triangle PSS' depends on which angle's opposite excircle is considered. The excenter opposite to ∠P traces the auxiliary circle (tangent at vertices where x = ±a), while the excenter opposite to focal angles traces a hyperbola with specific parametric equations h = ae·sec(θ)·(e+1)/(e-1), k = be·tan(θ).
For the locus of eccentric opposite to angle $S'$ (a focal angle), the equation is $h = ae\sec\theta \cdot \frac{e+1}{e-1}$ and $k = be\tan\theta$ for $ae\sec\theta > 0$, which traces a hyperbola. For the eccentric opposite to angle $P$, simplifying the expression $h = \frac{-2a^2e\sec\theta + a^2e^2\sec\theta + a^2e - a^2e^2\sec\theta + a^2e}{2ae\sec\theta - 2ae}$ yields $h = -a$ for $ae\sec\theta 0$, making the locus tangent at the vertex.
Correct Answer: 1,2