Question

Find the value of $x$, if the two lines, through $a=\left(5,6\right)$ and $b=\left(2,-3\right)$ and through $c=\left(-3,3\right)$ and $d=\left(6,x\right)$ are perpendicular.

Answer 1

Step 1: Find the slopes of the lines

Given: $a\left(5,6\right),b\left(2,-3\right),c\left(-3,3\right),d\left(6,x\right)$

We know that when lines are perpendicular, the product of slopes: $-1$

The slope of $ab$,say $m_1=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}$

$$\begin{gathered} \Rightarrow m_1=\frac{-3-6}{2-5} \\ =\frac{-9}{-3} \\ =3 \end{gathered}$$

The slope of $cd$, say $m_2=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}$

$$\begin{gathered} \Rightarrow m_2=\frac{3-x}{-3-6} \\ =\frac{3-x}{-9} \\ =\frac{x-3}{9} \end{gathered}$$

Step 2: Check whether lines are perpendicular or not

Now, we know

$$\begin{gathered} m_1{m}_2=-1 \\ \Rightarrow 3\times \frac{x-3}{9}=-1 \\ \Rightarrow x-3=-3 \\ \Rightarrow x=0 \end{gathered}$$

Hence, $x=0$ is the answer.