Question

Find the equation of the circle with center at $(-3,5)$ and passes through the point $(5,-1)$

• A. ${(x+3)}^2+{(y-5)}^2=100$
• B. $(x-3{)}^2+(y-5{)}^2$
• C. $(x+3{)}^2+(y-5{)}^2$
• D. None of the above

Answer 1

The correct option is A ${(x+3)}^2+{(y-5)}^2=100$

Here we know that $(h,k)=(-3,5)$ but we are not given the radius.

However, we can find the radius by using the distance formula.

Therefore,

$r=\sqrt{(x_2–x_1{)}^2+(y_2–y_1{)}^2}$

$r=\sqrt{(5-(-3){)}^2+(-1-5{)}^2}$

$r=\sqrt{(8{)}^2+(6{)}^2}$

$r=\sqrt{64+36}$

$r=\sqrt{100}$

$r=10$

An equation of the circle with center $(h,k)$ and radius $r$ is

$(x-h{)}^2+(y-k{)}^2=r^2$

Substitute the values in the equation of circle as follows:

$(x-(-3){)}^2+(y-5{)}^2={10}^2$

i.e., $(x+3{)}^2+(y-5{)}^2=100$

Therefore, the equation of the circle is $(x+3{)}^2+(y-5{)}^2=100$.