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.
The number of points on the hyperbola and the circle from which tangents drawn to the circle and the hyperbola, respectively, are perpendicular to each other is

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`.