How to get the equation of a circle with the given information

Question

You have a center of a circle $M(-2,3)$ going through a point $P(1,7)$. What is the equation of the circle?

I thought you could solve it by $R^2 = 3^2 + 4^2$, but that would just give a radius of 5 because there aren't any $x$ or $y$'s. Then I thought, should you then add a point $P(x,y)$? I'm not too sure, can someone give me hints?

Answer 1

You correctly calculated the radius, now you just need to shift the circle into position. The general equation is

$r^2 = (x-x_c)^2 + (y-y_c)^2$

Where the center is $(x_c,y_c).$ So your equation becomes

$r^2 = (x+2)^2 + (y-3)^2$

You do not need to add an extra point, since a circle is already determined by its center and an arbitraty point on the circle.

Answer 2

The equation of a circle, centre $(p,q)$ and radius $r$, is

$$ (x-p)^2 + (y-q)^2 = r^2 \, . $$

Here $p$, $q$ and $r$ will be fixed numbers and the circle is the set of points $(x,y)$ for which the equation holds, for example: $(x-1)^2 + (y+2)^2 = 4$ is the equation of the circle with centre $(1,-2)$ and radius $2$.

In your case, the centre is $(-2,3)$. There are lots of circles centred at $(-2,3)$. They have equations $(x+2)^2 + (y-3)^2 = r^2$ where $r > 0$ is the radius of the circle. You also know that the circle passes through the point $(1,7)$. We can use this to find the radius. The circle $(x+2)^2 + (y-3)^2 = r^2$ passes through the point $(1,7)$ if and only if $(x,y) = (1,7)$ is a solution to the equation $(x+2)^2 + (y-3)^2 = r^2$. Let's put $(x,y) = (1,7)$ into the equation:

$$(1+2)^2 + (7-3)^2 = r^2 \iff 3^2 + 4^2 = r^2 \iff 25 = r^2 \, . $$

Since $r>0$, we have $r = 5$ (instead of $r = -5$). It follows that your circle has the equation

$$(x+2)^2 + (y-3)^2 = 25 \, . $$

It is the circle centred at $(-2,3)$ with radius $5$.

Answer 3

You correctly figured out the distance from the center of the circle to point P on the circle. You did this through the calculation

$$r^2=[1-(-2)]^2+(7-3)^2$$

to obtain a radius of length 5. Now, from the definition of a circle, the distance from the center to each point $(x,y)$ on the circle is the same. This distance can be calculated in the exact same way.

$$[x-(-2)]^2+(y-3)^2=5^2\text{ or }(x+2)^2+(y-3)^2=25$$