Question

If the circle $x^{2}+y^{2}-6x-12y-55=0$ intercepts the x-axis at two points $A$ and $B$, then $|AB|$ is equal to

• A. 6
• B. 8
• C. 10
• D. 12
• E. 16

Hint:

Logic Tip: The algebraic method (setting $y=0$) is often less prone to sign errors than extracting $g, f, c$ and using the formula, especially when the coefficient of $x^2$ and $y^2$ might not be 1.

Answer 1

Answer: (E)

Concept:
The length of the intercept made by the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ on the x-axis is given by the formula $2\sqrt{g^2 - c}$. Alternatively, the x-intercepts can be found by setting $y = 0$ and solving the resulting quadratic equation for $x$.
Step 1: Method 1: Use the intercept formula.
Compare the given circle equation $x^2 + y^2 - 6x - 12y - 55 = 0$ with the general form: $2g = -6 \implies g = -3$ $2f = -12 \implies f = -6$ $c = -55$ The length of the x-intercept is: $$|AB| = 2\sqrt{g^2 - c}$$ $$|AB| = 2\sqrt{(-3)^2 - (-55)}$$ $$|AB| = 2\sqrt{9 + 55} = 2\sqrt{64} = 2 \cdot 8 = 16$$
Step 2: Method 2: Solve algebraically (Alternative).
To find where the circle crosses the x-axis, set $y = 0$: $$x^2 + (0)^2 - 6x - 12(0) - 55 = 0$$ $$x^2 - 6x - 55 = 0$$ Factor the quadratic equation: $$(x - 11)(x + 5) = 0$$ The x-coordinates of the points A and B are $x = 11$ and $x = -5$. The distance between them is: $$|AB| = |11 - (-5)| = |11 + 5| = 16$$