Question

If the distance between $(2,3)$ and $(-5,2)$ is equal to the distance between $(x,2)$ and $(1,3)$, then the values of$x$ are

• A. $-6,8$
• B. $6,8$
• C. $-8,6$
• D. $-7,7$

Answer 1

The correct option is A

$-6,8$

Explanation for correct option

Given that the distance between $(2,3)$ and $(-5,2)$ is equal to the distance between $(x,2)$ and $(1,3)$

We know that the distance between two points say $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ $=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Distance between $(2,3)$ and $(-5,2)$ = $=\sqrt{{(-5-2)}^2+{(2-3)}^2}=\sqrt{49+1}=\sqrt{50}$

Distance between $(x,2)$and$(1,3)$ $=\sqrt{{(1-x)}^2+{(3-2)}^2}=\sqrt{1-2x+x^2+1}=\sqrt{2-2x+x^2}$

Since the distance are equal,

$$\begin{gathered} \therefore \sqrt{2-2x+x^2}=\sqrt{50} \\ \Rightarrow 2-2x+x^2=50 \\ \Rightarrow x^2-2x-48=0 \\ \Rightarrow x^2-8x+6x-48=0 \\ \Rightarrow x\left(x-8\right)+6\left(x-8\right)=0 \\ \Rightarrow \left(x-8\right)\left(x+6\right)=0 \\ \Rightarrow x=-6,8 \end{gathered}$$

Hence option (A) is correct i.e. $-6,8$