Question

The equations of the circle which pass through the origin and make intercepts of lengths $4$ and $8$ on the $x$ and $y$ - axes respectively are

• A. $x^2+y^2\pm 4x\pm 8y=0$
• B. $x^2+y^2\pm 2x\pm 4y=0$
• C. $x^2+y^2\pm 8x\pm 16y=0$
• D. $x^2+y^2\pm x\pm y=0$

Answer 1

The correct option is A

$x^2+y^2\pm 4x\pm 8y=0$

Estimate the Equation of the Circle:

Let $S=x^2+y^2+2gx+2fy=0$ be the equation of the circle which passes through the origin, $C=\left(-g,-f\right),r=\sqrt{g^2+f^2}$ .

Given that it makes intercept of $4$ with $x$-axis $\Rightarrow 4=2\sqrt{r^2-d^2}$.

where $d$ is distance between center and chord

$C$ is the center of the circle,

$g$ and $f$ are coefficients used in the general equation of the circle.

$$\begin{gathered} \Rightarrow 2=\sqrt{g^2+f^2-f^2} \\ \Rightarrow g=\pm 2 \\ \Rightarrow 4=\sqrt{g^2+f^2-g^2} \\ \Rightarrow f=\pm 4 \end{gathered}$$

So, the equation of circle is $x^2+y^2\pm 4x\pm 8y=0$.

Hence, the required equation is $x^2+y^2\pm 4x\pm 8y=0$ and therefore option $\left(A\right)$ is correct.