Question

If a circle passing through the points $(1, 5)$ and $(4,0)$ makes equal intercepts on coordinate axes and if its centre lies in the first quadrant, then $\sqrt{4g^{2}-c^{2}}=$

• A. 2
• B. 4
• C. 3
• D. 5

Hint:

Equal intercepts on both axes means $g^2 = f^2$. First quadrant center means $g$ and $f$ are both negative.

Answer 1

The Correct Option is C

Step 1: Recall equal intercepts.
A circle $x^2+y^2+2gx+2fy+c=0$ cuts the x-axis a length $2\sqrt{g^2-c}$ and the y-axis a length $2\sqrt{f^2-c}$. Equal intercepts means $g^2=f^2$.
Step 2: Use the first quadrant centre.
The centre is $(-g,-f)$. For it to be in the first quadrant both $-g>0$ and $-f>0$, so $g$ and $f$ are negative and equal: $g=f$.
Step 3: Put the two points into the circle.
Using $g=f$, the circle is $x^2+y^2+2gx+2gy+c=0$. For $(1,5)$: $1+25+2g+10g+c=0\Rightarrow 12g+c=-26$. For $(4,0)$: $16+8g+c=0\Rightarrow 8g+c=-16$.
Step 4: Solve the two equations.
Subtract: $4g=-10$, so $g=-\frac52$. Then $c=-16-8(-\frac52)=4$.
Step 5: Check the centre is correct.
Centre $=(-g,-g)=\left(\frac52,\frac52\right)$, which is indeed in the first quadrant. Good.
Step 6: Compute the asked value.
\[ \sqrt{4g^2-c^2}=\sqrt{4\cdot\tfrac{25}{4}-16}=\sqrt{25-16}=\sqrt9=3. \] \[ \boxed{3} \]