Chapter 9.5, Problem 14PSB

To determine

To show: Given points lie on the circle with center $(2,-1)$

Explanation of Solution

Given information:Here, we have given that points are $(7,11)$ , $(7,-13)$ , $(14,4)$ and center $(2,-1)$ .

Formula used: Distance formula: d = $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Proof: Let points be A $(7,11)$ , B $(7,-13)$ , C $(14,4)$ and center O $(2,-1)$ .

So, OA = $\sqrt{{(2-7)}^2+{(-1-11)}^2}$

  $\begin{array}{l}\Rightarrow \sqrt{{(-5)}^2+{(-12)}^2}\\ \Rightarrow \sqrt{25+144}\\ \Rightarrow \sqrt{169}\\ \Rightarrow 13\end{array}$

Now, OB = $\sqrt{{(2-7)}^2+{[-1-(-13)]}^2}$

  $\begin{array}{l}\Rightarrow \sqrt{{(-5)}^2+{(12)}^2}\\ \Rightarrow \sqrt{25+144}\\ \Rightarrow \sqrt{169}\\ \Rightarrow 13\end{array}$

Similarly, OC = $\sqrt{{(2-14)}^2+{(-1-4)}^2}$

  $\begin{array}{l}\Rightarrow \sqrt{{(-12)}^2+{(-5)}^2}\\ \Rightarrow \sqrt{144+25}\\ \Rightarrow \sqrt{169}\\ \Rightarrow 13\end{array}$

Now, As OA, OB and OC are at equal distances from the center,

  $\Rightarrow$ OA, OB and OC are radius.

  $\therefore$ A, B and C lie on circle.