If three distinct numbers a , b , c are in G.P. and the equations a x 2 + 2 b x + c = 0 and d x 2 + 2 e x + f = 0 have a common root, then which one of the following statements is correct?

Question

If three distinct numbers $a, b, c$ are in G.P. and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?

Smart Achievers · Accepted Answer

(4)

$Given a, b, c are in G.P.$

Let $a = a,$ $b = a r$ and $c = a r^{2}$ ...(i)

Now, consider $a x^{2} + 2 b x + c = 0$

$∴$ $a x^{2} + 2 a r x + a r^{2} = 0$ $[Using (i)]$

$\Rightarrow$ $x^{2} + 2 r x + r^{2} = 0 \Rightarrow {(x + r)}^{2} = 0 \Rightarrow x = - r$

Since, $x = - r$ is also a root of $d x^{2} + 2 e x + f = 0$

$∴$ $d r^{2} - 2 e r + f = 0$

$\Rightarrow$ $d (\frac{c}{a}) - 2 e (\frac{c}{b}) + f = 0$ $[∵ (- r) (- r) = \frac{c}{a} and \frac{a r^{2}}{a r} = \frac{c}{b}]$

$\Rightarrow \frac{d c}{a} + f = \frac{2 e c}{b}$

$\Rightarrow \frac{d}{a} + \frac{f}{c} = \frac{2 e}{b}$

$∴$ $\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in A.P.$

Solution

Q.
If three distinct numbers $a, b, c$ are in G.P. and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct

1 $d, e, f are in A.P.$

2 $\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in G.P.$

3 $d, e, f are in G.P.$

4 $\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in A.P.$

Want Us to Email you About Special Offers & Updates?

Solution

Q. If three distinct numbers a,b,c are in G.P. and the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root, then which one of the following statements is correct

1 d,e,f are in A.P.

2 da,eb,fc are in G.P.

3 d,e,f are in G.P.