Question:
The length of the intercept made by the circle $x^2 + y^2 - 10x + 4y + 9 = 0$ on the x-axis is:
The length of the intercept made by the circle $x^2 + y^2 - 10x + 4y + 9 = 0$ on the x-axis is:
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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.
Updated On: May 31, 2026
- $8$
- $6$
- $10$
- $12$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The length of the intercept made by a standard circle $x^2 + y^2 + 2gx + 2fy + c = 0$ on the x-axis is given by the formula $2\sqrt{g^2 - c}$.
Step 2: Meaning
We identify the parameters $g$ and $c$ from the given circle's equation $x^2 + y^2 - 10x + 4y + 9 = 0$.
Step 3: Analysis
Comparing with the standard equation: \[ 2g = -10 \implies g = -5 \] \[ c = 9 \] Calculating the length of the x-intercept: \[ \text{Length} = 2\sqrt{g^2 - c} = 2\sqrt{(-5)^2 - 9} = 2\sqrt{25 - 9} = 2\sqrt{16} = 2(4) = 8 \]
Step 4: Conclusion
The length of the intercept made by the circle on the x-axis is $8$ units. Final Answer: (A)
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