Question

In the coordinate plane, the points $F(-2,1),G(1,4)$ and $H(4,1)$ lie on a circle with center $P$. What are the coordinates of point $P$ ?

• A. $(0,0)$
• B. $(1,1)$
• C. $(1,2)$
• D. $(1,-2)$

Answer 1

The correct option is B $(1,1)$

Let $H\equiv (x_1,y_1)=(4,1)$

$G\equiv (x_2,y_2)=(1,4)$

$F\equiv (x_3,y_3)=(-2,1)$

$P\equiv (x,y)$

slope of $m_r=\frac{y_2-y_1}{x_2-x_1}=\frac{4-1}{1-4}=-1$

slope of $m_t=\frac{y_3-y_2}{x_3-x_2}=\frac{1-4}{-2-1}=1$

Formula for x-coordinate of the center of the circle is given as:

$x=\frac{m_r{m}_t(y_3-y_1)+m_r(x_2+x_3)-m_t(x_1+x_2)}{2(m_r-m_t)}$

$\therefore x=\frac{(-1)\left(1\right)(1-1)+-1(1+(-2\left)\right)-1(4+1)}{2(-1-1)}$

$\therefore x=\frac{0+1-5}{-4}=1$

$y=-\frac{1}{m_r}[x-\frac{x_1+x_2}{2}]+\frac{y_1+y_2}{2}$

$\therefore y=-\frac{1}{-1}[1-\frac{4+1}{2}]+\frac{1+4}{2}$

$\therefore y=[\frac{2-5}{2}]+\frac{5}{2}=1$

$\therefore P\equiv (1,1)$

Hence, option $B$ is correct.