MCQ
If the equation $(a^2+ b^2)x^2- 2(ac + bd)x + c^2+ d2 = 0$ has equal roots, then:
- A$\text{ab}=\text{cd}$
- ✓$\text{ad}=\text{bc}$
- C$\text{ad}=\sqrt{\text{bc}}$
- D$\text{ab}=\sqrt{\text{cd}}$
✓
Answer
Correct option: B.
$\text{ad}=\text{bc}$
In the equation
$ \Rightarrow\left(a^2+b^2\right) x^2-2(a c+b d) x+\left(c^2+d^2\right)=0 $
$ \Rightarrow D=B^2-4 A C $
$ \Rightarrow D=[-2(a c+b d)]^2-4\left(a^2+b^2\right)\left(c^2+d^2\right) $
$ \Rightarrow D=4\left[a^2 c^2+b 2 d^2+2 a b c d\right]-4\left[a^2 c^2+a^2 d^2+b^2 c^2+b^2 d^2\right] $
$ \Rightarrow D=4 a^2 c^2+4 b^2 d^2+8 a b c d-4 a^2 c^2-4 a^2 d^2-4 b^2 c^2-4 b^2 d^2 $
$ \Rightarrow D=8 a b c d-4 a^2 d 2-4 b^2 c^2 $
$ \Rightarrow D=-4\left[a^2 d^2+b^2 c^2-2 a b c d\right] $
$ \Rightarrow D=-4(a d-b c)^2$
$\because$ Roots are equal
$\therefore D = 0$
$\Rightarrow -4(ad - bc)^2= 0$
$\Rightarrow ad - bc = 0$
$\Rightarrow ad = bc$
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