Question

If $\frac{1}{ p }, \frac{1}{ q }, \frac{1}{ r }$ are in A.P. and $p$ and $r$ be different having same sign, then the roots of the equation $px ^2+2 qx + r =0$ will be

• A. real
• B. equal
• C. imaginary
• D. real and distinct

Answer 1

Answer: (C)

Since, $\frac{1}{p}, \frac{1}{q}, \frac{1}{r}$ in $A P \Rightarrow q=\frac{2 p r}{p+r}$
Now,
$D=4 q^2-4 p r =-4\left[p r-\left(\frac{2 p r}{p+r}\right)^2\right] $
$=-(p r)\left[2\left(\frac{p-r}{p+r}\right)\right]^2$
[From equation (i)]
Since, $pr >0, p \neq r$ given, $D \neq 0$ and $D <0$.
Hence, the roots are imaginary.