Question

lf the roots of the equation $x^2-2cx+$ $ab$ $=0$ be real and unequal, the roots of the equation $x^2-2(a+b)x+(a^2+b^2+2c^2)=0$ are

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

Answer 1

The correct option is D imaginary

It is given that the roots of
$x^2-2cx+ab=0$ are real.
Hence $D>0$
$B^2-4AC>0$
$4c^2-4ab>0$
$c^2>ab$ ...(i)
Consider the second equation
$x^2-2(a+b)x+(a^2+b^2+2c^2)=0$
$D$$=B^2-4AC$
$=4(a+b{)}^2-4(a^2+b^2+2c^2)$
$=4(a^2+b^2+2ab)-4(a^2+b^2+2c^2)$
$=8ab-8c^2$
$=8(ab-c^2)$
Now from i
$c^2>ab$
$ab-c^2<0$
hence
$8(ab-c^2)<0$
Hence the given equation has imaginary roots.