# Thread: Equation of the Line & Two Circles

1. ## Equation of the Line & Two Circles

Find an equation of the line containing the centers of the two circles given below.

x^2 + y^2 - 4x + 6y + 4 = 0 and

x^2 + y^2 + 6x + 4y + 9n= 0.

Seeking the first-two steps.

2. ## Re: Equation of the Line & Two Circles

1.) Put both circles in the form:

$\displaystyle (x-h)^2+(y-k)^2=r^2$

2.) Use the formula:

$\displaystyle y=\frac{y_2-y_1}{x_2-x_1}(x-x_1)+y_1$

3. ## Re: Equation of the Line & Two Circles

Originally Posted by MarkFL
1.) Put both circles in the form:

$\displaystyle (x-h)^2+(y-k)^2=r^2$

2.) Use the formula:

$\displaystyle y=\frac{y_2-y_1}{x_2-x_1}(x-x_1)+y_1$
I understand step 1. Are the values for x_1 & y_1 needed for step 2 found in your first step?

4. ## Re: Equation of the Line & Two Circles

Originally Posted by harpazo
I understand step 1. Are the values for x_1 & y_1 needed for step 2 found in your first step?
The second formula MarkFL gave you is used to find the equation of the line through the two points (x1, y1) and (x2, y2).

You are required to find the equation of the line through the two circle centres - so what are those two points?

5. ## Re: Equation of the Line & Two Circles

Originally Posted by harpazo
I understand step 1. Are the values for x_1 & y_1 needed for step 2 found in your first step?
The first equation is (x - 2)^2 + (y + 3)^2 = 25. The center (h, k) is (2, -3).

The second equation is (x + 3)^2 + (y + 2)^2 = 16. The center (h, k) is (-3, -2).

Let x_1 = 2, y_1 = -3, x_2 = -3 and y_2 = -2.

We now plug and chug.

y = [(-2 -(-3))/(-3 -2)](x - 2) - 3

y = [(-2 + 3)/-5](x - 2) - 3

y = (-1/5)(x - 2) - 3

y = [-(x - 2)/5] - 3

Correct?

6. ## Re: Equation of the Line & Two Circles

Originally Posted by Debsta
The second formula MarkFL gave you is used to find the equation of the line through the two points (x1, y1) and (x2, y2).

You are required to find the equation of the line through the two circle centres - so what are those two points?

7. ## Re: Equation of the Line & Two Circles

Originally Posted by harpazo
The first equation is (x - 2)^2 + (y + 3)^2 = 25. … this should be 9.The center (h, k) is (2, -3). Centre is correct.

The second equation is (x + 3)^2 + (y + 2)^2 = 16 … this should be 4. The center (h, k) is (-3, -2). Centre is correct.

Let x_1 = 2, y_1 = -3, x_2 = -3 and y_2 = -2.

We now plug and chug.

y = [(-2 -(-3))/(-3 -2)](x - 2) - 3

y = [(-2 + 3)/-5](x - 2) - 3

y = (-1/5)(x - 2) - 3

y = [-(x - 2)/5] - 3

Correct?
Final equation is correct. You are not getting the centre/radius from of the circle correct though.

Equations of lines are usually written in:

gradient-intercept form: y=mx + c , in this case $\displaystyle y=\frac{-1}{5}x - \frac{13}{5}$

or

standard form: ax+ by +c =0 , in this case x + 5y + 13 =0

8. ## Re: Equation of the Line & Two Circles

Originally Posted by Debsta
Final equation is correct. You are not getting the centre/radius from of the circle correct though.

Equations of lines are usually written in:

gradient-intercept form: y=mx + c , in this case $\displaystyle y=\frac{-1}{5}x - \frac{13}{5}$

or

standard form: ax+ by +c =0 , in this case x + 5y + 13 =0
Thank you for pointing out my error(s) in red. This is perfect for my notes. I will practice more and more as I go through the Michael Sullivan textbook.