If one root of the quadratic equation $px^2 + qx + r = 0 (p \neq 0)$ is a surd $\displaystyle \frac{\sqrt{a}}{\sqrt{a} + \sqrt{a-b}}$, where $p, q, r, a, b$ are all rationals, then the other root is
-
$\displaystyle \frac{\sqrt{b}}{\sqrt{a} - \sqrt{a-b}}$
-
$\displaystyle a + \frac{\sqrt{a(a-b)}}{b}$
-
$\displaystyle \frac{a + \sqrt{a(a-b)}}{b}$
-
$\displaystyle \frac{\sqrt{a} - \sqrt{a-b}}{\sqrt{b}}$
C
Correct answer
Explanation
The given root is sqrt(a)/(sqrt(a) + sqrt(a-b)). Rationalizing the denominator by multiplying by (sqrt(a) - sqrt(a-b)) gives (sqrt(a^2) - sqrt(a(a-b))) / (a - (a-b)) = (a - sqrt(a(a-b))) / b. Since the coefficients are rational, the conjugate root must be (a + sqrt(a(a-b))) / b.