<p>If $(a, 0)$ is a point on a diameter of the circle $x^2 + y^2 = 4$, then $x^2 - 4x - a^2 = 0$ must have:</p>
<p>(a) exactly one real root in $\left[-\frac{9}{10}, \frac{1}{10}\right]$</p>
<p>(b) exactly one real root in $\left[4, \frac{49}{10}\right]$</p>
<p>(c) exactly one real root in $[0, 2]$</p>
<p>(d) two distinct real roots in $[-1, 5]$</p>
Step-by-Step Solution
Key Concept: Determine the constraint on $a$ from the diameter condition, find the roots using the quadratic formula, and verify which interval contains exactly one root.
<p><strong>Analysis:</strong> Since $(a, 0)$ lies on a diameter of circle $x^2 + y^2 = 4$, we have $a^2 \leq 4$, so $-2 \leq a \leq 2$. The equation $x^2 - 4x - a^2 = 0$ has roots $x = 2 \pm \sqrt{4 + a^2}$. Since $a^2 \in [0,4]$, we have $4 + a^2 \in [4, 8]$, so $\sqrt{4+a^2} \in [2, 2\sqrt{2}]$. The roots are $x_1 = 2 - \sqrt{4+a^2} \in [2-2\sqrt{2}, 0] \approx [-0.83, 0]$ and $x_2 = 2 + \sqrt{4+a^2} \in [4, 2+2\sqrt{2}] \approx [4, 4.83]$. Checking the interval $\left[4, \frac{49}{10}\right] = [4, 4.9]$: $x_2 \in [4, 4.83] \subset [4, 4.9]$, so exactly one root lies in this interval.</p><p>∴ Answer is (b).</p>
Correct Answer: b