Question

The length of the intercept made by the circle $x^2 + y^2 - 10x + 4y + 9 = 0$ on the x-axis is:

• A. $8$
• B. $6$
• C. $10$
• D. $12$

Hint:

The x-intercept of a circle is $2\sqrt{g^2 - c}$, while the y-intercept is $2\sqrt{f^2 - c}$. Memorizing both helps solve intercept problems instantly.

Answer 1

The Correct Option is A

Step 1: Recall the x-intercept rule.
The chord that a circle cuts on the x-axis has length $2\sqrt{g^2 - c}$, where $g$ and $c$ come from the general circle $x^2 + y^2 + 2gx + 2fy + c = 0$.

Step 2: Compare with the general form.
Our circle is $x^2 + y^2 - 10x + 4y + 9 = 0$. Match the $x$ term: $2g = -10$, so $g = -5$. The constant gives $c = 9$.

Step 3: Note we do not need $f$.
The x-axis intercept only uses $g$ and $c$. The $f$ value would matter for the y-axis instead.

Step 4: Compute $g^2 - c$.
Square $g$ and subtract $c$.
\[ g^2 - c = (-5)^2 - 9 = 25 - 9 = 16 \]

Step 5: Take the square root and double.
The length is twice the square root.
\[ 2\sqrt{16} = 2\times 4 = 8 \]

Step 6: State the answer.
So the circle cuts a chord of length $8$ on the x-axis.
\[ \boxed{8} \]