Question

If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$. The equation with roots as $k\alpha \&k\beta \forall k\in R$ is:

• A. $a(x-k{)}^2+b(x-k)+c=0$
• B. $a(kx{)}^2+b(kx)+c=0$
• C. $ka{x}^2+bkx+c=0$
• D. $a{\left(\frac{x}{k}\right)}^2+b\left(\frac{x}{k}\right)+c=0$

Answer 1

The correct option is D $a{\left(\frac{x}{k}\right)}^2+b\left(\frac{x}{k}\right)+c=0$

Given: $a{x}^2+bx+c=0,a\ne 0$ with roots $\alpha ,\beta$
To find: Transformed Quadratic equation with roots $k\alpha$ and $k\beta$

Let $t=kx$

Since, $x=\alpha \Rightarrow t=k\alpha$
$x=\beta \Rightarrow t=k\beta$

Replacing $x\to \frac{t}{k},$ we get the equation as:
$a{\left(\frac{t}{k}\right)}^2+b\left(\frac{t}{k}\right)+c=0$

Now in terms of variable $x,$ the equation becomes:

$a{\left(\frac{x}{k}\right)}^2+b\left(\frac{x}{k}\right)+c=0$
which is the required equation.