# coordinate geometry

#### thorpelizts

##### New member
Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.

how do i even begin?

#### Wilmer

##### In Memoriam
Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.
how do i even begin?
Teacher gave no instructions, no teaching?

#### Sudharaka

##### Well-known member
MHB Math Helper
Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.

how do i even begin?
Hi thorpelizts,

Let $$P\equiv (-3,4)$$ and let $$Q$$ be the point of intersection of the circle and the y-axis. Since the y-axis is a tangent to the circle, $$PQ$$ is perpendicular to the y-axis. Now I am sure you can find the length of $$PQ$$ which is the radius of the circle. Can you give it a try?

Kind Regards,
Sudharaka.

#### soroban

##### Well-known member
Hello, thorpelizts!

Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.

How do i even begin? . Make a sketch!

You are expected to know this formula: .$$(x-h)^2 + (y-k)^2 \:=\:r^2$$
. . where $$(h,k)$$ is the center and $$r$$ is the radius.

Code:
                            |
* * *       |
*           *   |
*               * |
*                 *|
|
*              r    *
*         * - - - - *4
*      (-3,4)       *
|
*                 *|
*               * |
*           *   |
* * *       |
|
- - - - - - - + - - - - + - - -
-3         |
You know $$h = -3,\;k=4.$$

Can you guess what the radius is?

#### MarkFL

Staff member
While soroban's method is easiest, you might also consider we want the solution of the systerm:

(x + 3)2 + (y - 4)2 = r2

x = 0

to have one real root.

Substitute into the first equation from the second:

(0 + 3)2 + (y - 4)2 = r2

(y - 4)2 + 9 - r2 = 0

We want this quadratic to have one root, hence the discriminant must be zero:

02 - 4(1)(9 - r2) = 0

r = 3

#### CaptainBlack

##### Well-known member
Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.

how do i even begin?
You begin by drawing a picture.

CB