JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Mock Test - Complex Numbers and Quadratic Equations

If \[a{{(p+q)}^{2}}+2bpq+c=0\,\,and\,\,a{{(p+r)}^{2}}+2bpr+c=0\,\,(a\ne 0),\] then

A) \[qr={{p}^{2}}\]

B) \[qr={{p}^{2}}+\frac{c}{a}\]

C) \[qr=-{{p}^{2}}\]

D) none of these

Correct Answer: B

Solution :

[b] Given, \[a{{(p+q)}^{2}}+2bpq+c=0\]and \[a{{(p+r)}^{2}}+2bpr+c=0\]

\[\Rightarrow q\]and r satisfy the equation \[a{{(p+x)}^{2}}+2bpx+c=0\]

\[\Rightarrow q\]and r are the roots of \[a{{x}^{2}}+2(ap+bp)x+c+a{{p}^{2}}=0\]

\[\Rightarrow qr=\]product of roots \[=\frac{c+a{{p}^{2}}}{a}={{p}^{2}}+\frac{c}{a}\]