Question

If the lines joining the points $(K,2),(3,6)$ and $(2,3),(K,7)$ are perpendicular to each other, then the possible value of $K$ is

• A. $\frac{5+\sqrt{65}}{2}$
• B. $\frac{5-\sqrt{65}}{2}$
• C. $\frac{5-\sqrt{65}}{4}$
• D. $\frac{5+\sqrt{65}}{4}$

Answer 1

Answer: (A), (B)

The correct options are
A $\frac{5+\sqrt{65}}{2}$
B $\frac{5-\sqrt{65}}{2}$

$m_1=$ Slope of line joining $(K,2)$ and $(3,6)$
$=\frac{6-2}{3-K}=\frac{4}{3-K}$
$m_2=$ Slope of line joining $(2,3)$ and $(K,7)$
$=\frac{7-3}{K-2}=\frac{4}{K-2}$

As the lines are perpendicular,
so, $m_1{m}_2=-1$
$\Rightarrow \frac{4}{3-K}\times \frac{4}{K-2}=-1$
$\Rightarrow -16=(3-K)(K-2)$
$\Rightarrow K^2-5K-10=0$
$\Rightarrow K=\frac{5\pm \sqrt{65}}{2}$