Question

The area of the quadrilateral formed by the tangent from the point $(4,5)$ to the circle $x^2+y^2-4x-2y-c=0$ with a pair of radii joining the points of contacts of these tangents is $8$ sq. units. The value of $c$ is

• A. $12$
• B. $-1$
• C. $3$
• D. $11$

Answer 1

Answer: (B), (D)

$11$
$-1$

Given equation of circle is $x^2+y^2-4x-2y-c=0$

$(-g,-f)=(2,1)$

Radius $=\sqrt{g^2+f^2-c}$, $h$ of sub tangent

Length of subtangent from point $(x_1,y_1)=\sqrt{x_1^2+y_1^2-4x_1-2y_1-c}$

Area of quadrilateral = length of subtangent x radius

$\Rightarrow \sqrt{4^2+5^2-4\times 4-2\times 5-c}\times \sqrt{4+1+c}$

$\Rightarrow 8^2=(15-c)(5+c)$

$\Rightarrow c^2-10c-11=0$

$\Rightarrow c=11,-1$