Question

The values of $k$, for which the equations $x^2-kx-21=0$ and $x^2-3kx+35=0$will have a common roots, is

• A. $k=\pm 4$
• B. $k=\pm 1$
• C. $k=\pm 3$
• D. $k=0$

Answer 1

The correct option is A

$k=\pm 4$

Explanation for the correct option:

Given: $x^2-kx-21=0$ and $x^2-3kx+35=0$

Let $a$ be the common root of given equations.

$$\begin{gathered} a^2-ka-21=0-\left(1\right) \\ {a}^2-3ka+35=0-\left(2\right) \end{gathered}$$

Solve the equations $\left(1\right)$ and $\left(2\right)$

$\begin{array}{crcl}\Rightarrow & ka+21& =& 3ka-35\\ \Rightarrow & 21+35& =& 3ka-ka\\ \Rightarrow & 56& =& 2ka\\ \Rightarrow & \frac{56}{2k}& =& a\\ \Rightarrow & \frac{28}{k}& =& a\end{array}$

Put the value of $\mathrm{a}$ in an equation

$$\begin{gathered} a^2-ka-21=0 \\ \begin{array}{crcl}\Rightarrow & \frac{{28}^2}{k^{20}}-k\frac{28}{k}-21& =& 0\\ \Rightarrow & \frac{{28}^2}{k^2}-28-21& =& 0\\ \Rightarrow & \frac{784}{k^2}& =& 49\\ \Rightarrow & \frac{784}{49}& =& k^2\\ \Rightarrow & 16& =& k^2\\ \Rightarrow & k& =& \pm 4\end{array} \end{gathered}$$

Hence option A is the correct answer.