Question

Write the equation of the circle graphed below.

Answer 1

The equation of the circle is
`{ \left(x - 1 \right)^2 + \left( y - 2 \right)^2 = ( 25 )/( 4 ) }`.

Explanation:

This is the general standard equation for the circle centered at (h, k) with radius r.

`(x-h)^2+(y-k)^2=r^2`

From the graph we know that the center, the red point, is (1, 2) and the circle pass through the point (2.5, 4) the blue point.

To find the standard equation of the circle you must:

Step 1: Find circle radius.

To find a radius of a circle we will compute the distance between points
`C=(1, 2)` and
`P=(2.5, 4)={ \left( ( 5 )/( 2 ) , 4 \right) }`. The distance can be computed by using formula:

`r = √( \left(C_x - P_x\right)^2 + \left(C_y - P_y\right)^2 )\\\\r = \sqrt{ \left(1 - (5)/(2) )^2 + \left(2 - 4)^2 }`

`r=\sqrt{(3^2)/(2^2)+2^2}=\sqrt{(9)/(4)+4}=\sqrt{(25)/(4)}=(5)/(2)`

Step 2: Substitute the values of the radius and the center into the general standard equation for the circle.

`\begin{aligned} \left(x - 1 \right)^2 + \left(y - 2 \right)^2 &= \left(( 5 )/( 2 )\right)^2 \\ \left(x - 1 \right)^2 + \left( y - 2 \right)^2 &= ( 25 )/( 4 ) \end{aligned}`