Consider the hyperbola `(X^(2))/(9)-(y^(2))/(a^(2))=1` and the circle `x^(2)+(y-3)=9`.
Also, the given hyperbola and the ellipse `(x^(2))/(41)+(y^(2))/(16)=1` are orthogonal to each other.
Combined equation of pair of common tangents between the hyperbola and the circle is given be

Options

A. `x^(2)-y^(2)=0`
B. `x^(2)-9=0`
C. `9y^(2)-19x^(2)=0`
D. No common tangent.

Answer

B

Explanation

Ellipse and hyperbola are orthogonal and therefore, they are confocal.
`"So, "a_(h)e_(h)=a_(e)e_(e)`
`rArr" "a^(2)+9=41-16`
`rArr" "9+a^(2)=25`
`rArr" "a^(2)=16`
Thus, hyperbola is `(x^(2))/(9)-(y^(2))/(16)=1.`
So, common tangents to the circle and hyperbola are `x = pm 3`.
Director circle of hyperbola does not exist as `a lt b.`
Director circle of circle is
`x^(2)+(y-3)^(2)=18`
`rArr" "x^(2)+y^(2)-6y-9=0`
This meets the hyperbola `16x^(2)-9y^(2)=144` at four points from where tangents drawn to the circle `x^(2)+(y-3)^(2)=9` are perpendicular to each other.
Let midpoint of AB be (h,k).
So, equation of line AB is `hx+ky-3(y+k)=h^(2)+k^(2)-6k`.
Since tangents at C and D intersect at the directrix, CD is the focal chord of hyperbola.
So, AB passes through focus of the hyperbola and that is `(pm5,0)`.
Therefore, required lacus is `x^(2)+y^(2)pm5x-3y=0`.