CircleHardBloom L3
Question
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:
Options
A.exactly one real root in $\left\lbrack - \frac{9}{10},\frac{1}{10} \right\rbrack$
B.exactly one real root in $\left\lbrack 4,\frac{49}{10} \right\rbrack$
C.exactly one real root in $\lbrack 0,2\rbrack$
D.two distinct real roots in $\lbrack - 1,5\rbrack$
Solution
Sol. [Key Concept: $(a,0)$ on a diameter of $x^{2} + y^{2} = 4$ means $|a| \leq \ 2$; analyse the roots of $x^{2} - 4x - a^{2} = 0$ on the stated intervals via its discriminant.]
Exactly one root in
$$\lbrack 2 - 2\sqrt{2},0\rbrack \cup \lbrack 4,2 + 2\sqrt{2}\rbrack$$
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