Question

Find an equation of the circle and sketch it if it has:

Answer 1

Answer:

$(x+2)^2+(y-4)^2=16$

Step-by-step explanation:

OK, lets start by drawing a basic graph (the first one) so we can visualize.

We already know that the y coordinate of the circle's center is $4$.

We know that the circle is tangent to the $X$ axis at $(-2,0)$

That means the x coordinate of the center has to be $-2$, as the tangent is a point on the edge of the circle that touches a line at exactly one point.

The radius is the distance from the center of the circle to its edge. We know the center's location now, it is $(-2, 4)$ and a point on the edge of the circle (the tangent point) which is $(-2,0)$. so the distance between the points is 4 which is the radius (you can use the distance formula, but it's quite oblivious.)

We can imagine the circle should look like this (the second one):

Now we can piece together an equation

The equation of a circle is $(x-h)^2+(y-k)^2=r^2$ where $(h,k)$ is the center and $r$ is the radius. When we put the numbers in: we get $(x-(-2))^2+(y-(4))^2=4^2$ which can be simplified into $(x+2)^2+(y-4)^2=16$ which is the answer.