Question

If $(a,0)$ is a point on a diameter of the circle $x^2+y^2=4$, then $x^2-4x-a^2=0$ has

• A. Exactly one real root in $(-2,1)$
• B. Exactly one real root in $[2,5]$
• C. Distinct roots greater than $1$
• D. Distinct roots less than $-1$

Answer 1

The correct option is A Exactly one real root in $(-2,1)$

Given:$X^2-4x-a^2=0$

Discriminant D=$(-4{)}^2-4(1\left)\right(-a^2)⟹16+4a^2$

Sum of roots =$4$

product of roots =$-a^2$

That means one root is positive and other is negative

$⟹-2\le x\le 2$