Consider a hyperbola H : `x^(2)-y^(2)` =k and a parabola `P:y=x^(2)` then identify the correct statements(S)

Options

A. If point of intrsections of P and H are concyclic then `k lt 2`
B. If P and H touch each other then `k = 1//4`
C. If `k=-1//3` and `m_(1)`, are the slopes of common tangents to P and H then `(3m_(1).^(2)+8)(3m_(2).^(2)+8)=112`
D. If P,H do not touch but intersect at exactly two points then `k lt 0`

Answer

A, B, C, D

Explanation

`y-y^(2)=k`
for concyclic `D ge 0` ge
for common tangent
`y=mx pm sqrt(km^(2)-k)` and `y = mx-(1)/(4)m^(2)`
comparing `(3m_(1).^(2)+8)(3m_(2).^(2)+8)=112`