Question

If the roots of the equation $\left({\text{a}}^{\text{2}}\text{+}{\text{b}}^{\text{2}}\right){\text{x}}^{\text{2}}\text{+ 2}\left(\text{bc + ad}\right)\text{x +}\left({\text{c}}^{\text{2}}\text{+}{\text{d}}^{\text{2}}\right)\text{= 0}$ are real then $a^2,bd,c^2$ are in

• A. AP
• B. GP
• C. HP
• D. AGP

Answer 1

Answer: (B)

GP

Given equation $(a^2+b^2)x^2+2(bc+ad)x+(c^2+d^2)=0$

roots are real

$D=b^2-4ac$

$=4(bc+ad{)}^2-4(a^2+b^2\left)\right(c^2+d^2)=0$

$=b^2{c}^2+a^2{d}^2+4abcd-(a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2)=0$

$=-a^2{c}^2-b^2{d}^2+4abcd=0$

$=(ac-bd{)}^2=0$

$\Rightarrow ac=bd$

$\Rightarrow a^2,bd,c^2$ are in G.P