Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# If one root of the equation, $3{{x}^{2}}-9x=kx-k$ is 2, then the value of k is: -(a) 4(b) 3(c) -6(d) -8

Last updated date: 17th Jun 2024
Total views: 403.2k
Views today: 12.03k
Verified
403.2k+ views
Hint: First of all take all the terms of the given quadratic equation to the left so that we can get 0 in the right hand side. Now, assume the two roots of the given quadratic equation as $\alpha$ and $\beta$ with $\alpha =2$. Apply the identity: - $\alpha +\beta =\dfrac{-b}{a}$ and $\alpha \beta =\dfrac{c}{a}$ to form two linear equation in k and $\beta$ and find the value of k. Remember that b = co-efficient of x and c = constant term while a = co – efficient of ${{x}^{2}}$.

Complete step-by-step solution
We have been given the quadratic equation: -
$\Rightarrow$$3{{x}^{2}}-9x=kx-k$
Taking all the terms to the L.H.S, we get,
\begin{align} & \Rightarrow 3{{x}^{2}}-9x-kx+k=0 \\ & \Rightarrow 3{{x}^{2}}-x\left( 9+k \right)+k=0 \\ \end{align}
Let us assume the two roots of the above equation as $\alpha$ and $\beta$. We have provided with one root equal to 2. Let $\alpha =2$. Here, we are assuming the coefficient of ${{x}^{2}}$, x, and the constant term is denoted by a, b and c respectively.
$\Rightarrow a=3,b=-\left( 9+k \right)$ and $c = k.$
Now, we know that sum of the roots of a quadratic equation is the ratio of (-b) and (a) and product of the roots is the ratio of (c) and (a).
\begin{align} & \Rightarrow \alpha +\beta =\dfrac{-b}{a} \\ & \Rightarrow 2+\beta =\dfrac{9+k}{3} \\ & \Rightarrow 6+3\beta =9+k \\ \end{align}
$\Rightarrow 3\beta -k=3$ ----- (i)
Also, we have,
\begin{align} & \Rightarrow \alpha \beta =\dfrac{c}{a} \\ & \Rightarrow 2\beta =\dfrac{k}{3} \\ \end{align}
$\Rightarrow \beta =\dfrac{k}{6}$ ------ (ii)
Substituting the value of $\beta$ from equation (ii) in equation (i), we have,
\begin{align} & \Rightarrow 3\times \dfrac{k}{6}-k=3 \\ & \Rightarrow \dfrac{k}{2}-k=3 \\ & \Rightarrow -\dfrac{k}{2}=3 \\ & \Rightarrow k=-6 \\ \end{align}
Hence, option (c) is the correct answer.

Note: You may note that we have assumed $\alpha =2$. You may assume, $\beta =2$, this will not affect the value of k in the answer. The only difference it will make is that at last, we have to eliminate $\alpha$ instead of $\beta$ just like we did above. There can be another way to solve the question if options are given. We will substitute the value of k one – by – one and find the roots. The option which will give one of the roots as 2 will be the answer.