Question

The locus of the centre of the circle $x^2 + y^2 + 4x \,\cos \,\theta - 2y\, sin \theta - 10 = 0$ is

• A. an ellipse
• B. a circle
• C. a hyperbola
• D. a parabola

Answer 1

Answer: (A)

an ellipse

Answer 2

The correct answer is A:an ellipse
If $\left(\alpha, \beta\right)$ is the centre, then $\alpha = -2\,cos\,\theta, \beta = sin \,\theta$
$\therefore\left(\frac{\alpha}{-2}\right)^{2}+\beta^{2} = cos^{2}\theta +sin^{2}\theta = 1$
$\Rightarrow \frac{\alpha^{2}}{4}+\beta^{2} = 1$
$\therefore$ locus of $\left(\alpha, \beta\right)$ is $\frac{x^{2}}{4}+\frac{y^{2}}{1}= 1$, an ellipse .