Question

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

• A. $x^2+y^2\pm4x\pm8y=0$
• B. $x^2+y^2\pm2x\pm4y=0$
• C. $x^2+y^2\pm8x\pm16y=0$
• D. $x^2+y^2\pm x\pm y=0$

Hint:

For a circle through the origin, $x$-intercept is $2g$ and $y$-intercept is $2f$.

Answer 1

The Correct Option is A

Step 1: Circle Through Origin
Equation: $x^2 + y^2 + 2gx + 2fy = 0$.
Step 2: Intercepts
x-intercept length $= |2g| = 4 \Rightarrow g = \pm 2$.
y-intercept length $= |2f| = 8 \Rightarrow f = \pm 4$.
Step 3: Center
The center is $(-g, -f) = (\pm 2, \pm 4)$.
Step 4: Final Equations
Substitute $g$ and $f$ values:
$x^2 + y^2 \pm 4x \pm 8y = 0$.
Final Answer: (a)