Question

If the area of the circle $ 7x^2 + 7y^2 - 7x + 14y + k = 0$ is $12 \: \pi \, sq$. units, then the value of k is

• A. $ \frac{- 43}{4}$
• B. $ \frac{- 301}{4}$
• C. $ - 16 $
• D. $\pm 4 $

Answer 1

Answer: (B)

Equation of circle
$ 7x^2 + 7y^2 - 7x + 14y + k = 0$
can be written as.
$x^2 + y^2 - x + 2y + \frac{k}{7} = 0 $
$ \therefore $ Centre of circle is $\left(\frac{1}{2} , -1\right)$
Radius of circle, $ r=\sqrt{\frac{1}{4} + 1- \frac{k}{7}} $
$= \sqrt{\frac{5}{4} - \frac{k}{7}} \Rightarrow r^{2} = \frac{5}{4} - \frac{k}{7} $
Given that area of circle = 12$\pi $ sq. units
$\Rightarrow \pi r^{2} = \pi\left(\frac{5}{4} - \frac{k}{7}\right) = 12\pi $
$\Rightarrow \frac{5}{4} - \frac{k}{7} = 12 $
$\Rightarrow \frac{5}{4} - 12 = \frac{k}{7} \Rightarrow \frac{-43}{4} = \frac{k}{7} $
$\Rightarrow k= \frac{-301}{ 4 }$