Circles
Circle
nta_abhyas_2025
Grade 11
Question:
Let $A(0, 3)$ and $B(0, 12)$ be two vertices of a $\triangle ABC$ where $C = (x, 0)$. If the circumcircle of $\triangle ABC$ touches the $z$-axis, then the value of $\cos 2\theta$ is (where, $\theta$ is angle $ACB$)
Step-by-Step Solution
Key Concept: Equal chords from a point and the geometric constraint of the circle equation determine the coordinates and the value of $x$
From the given conditions $OA = OB = OC^2$ and the constraint $3 + 12 = x^2$, we solve for $x$: $x^2 = 15$ leads to $x = 6$ (taking the positive value). The center of the circle is located at $\left(6, \frac{15}{2}\right)$, confirming that $x = 6$.
Correct Answer: 6