Equation of Circle in Standard Form

[mathdad]

Determine the equation of the circle that passes through the point (-4, 1) and its center is the midpoint of the line segment joining the centers of the two circles given below:

x^2 + y^2 -6x - 4y + 12 = 0

and

x^2 + y^2 - 14x + 47 = 0.

Write the equation in standard form.

1. Does the question involve completing the square on the two circles given?

2. After finding the center for both circles in the form
(h, k), must I use the midpoint formula?

3. What other hints can you give to guide me through this question?

 

[MarkFL]

Determine the equation of the circle that passes through the point (-4, 1) and its center is the midpoint of the line segment joining the centers of the two circles given below:

x^2 + y^2 - 4y + 12 = 0

and

x^2 + y^2 - 14x + 47 = 0.

Write the equation in standard form.

1. Does the question involve completing the square on the two circles given?

Yes, standard form is:

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

where the center is the point $(h,k)$ and the radius is $r$.

2. After finding the center for both circles in the form
(h, k), must I use the midpoint formula?

Yes, once you determine the centers of the two given circles, then the midpoint of the line segment joining those two center points will be the center of the circle you are asked to find.

3. What other hints can you give to guide me through this question?

Once you have determine the center of the circle, then you want to find the distance from the center to the given point said to be on the circle, and that is your radius. Once you have the radius, along with the previously found center, you have all you need to write the equation. :)

 

[mathdad]

Yes, standard form is:

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

where the center is the point $(h,k)$ and the radius is $r$.
Yes, once you determine the centers of the two given circles, then the midpoint of the line segment joining those two center points will be the center of the circle you are asked to find.
Once you have determine the center of the circle, then you want to find the distance from the center to the given point said to be on the circle, and that is your radius. Once you have the radius, along with the previously found center, you have all you need to write the equation. :)

Cool. I want to ask you for a math favor. As I post problems almost everyday, can you provide the steps needed to solve each problem? For example, step 1 do this, step 2 do that, etc. This will allow me to find the answer on my own.

This is also good practice in terms of following directions or instructions. Lastly for today, I have 5 personal questions unrelated to mathematics to ask you. Is this ok?

I can PM all 5 questions later today. I really would like your opinion. You do have to rush to answer my personal questions. I just want to know what you think. One of the questions involves the Navy. Look for my PM questions later today.

P. S. I will answer this question concerning two circles tomorrow. I am off tomorrow. Trust me, you are not wasting your time with me. I follow your instructions as I try to increase my math solving skills. In fact, I have learned so much from your replies since joining the MHB.

 

[MarkFL]

Cool. I want to ask you for a math favor. As I post problems almost everyday, can you provide the steps needed to solve each problem? For example, step 1 do this, step 2 do that, etc. This will allow me to find the answer on my own.

Providing complete step by step instructions for completing a problem generally isn't necessary unless the OP has no idea how to begin the problem...and even then it would really depend on the situation. That could be a great deal of work to do as standard practice. I would rather hear from the OP what they think the steps should be, and then I can offer any advice based on that.

This is also good practice in terms of following directions or instructions. Lastly for today, I have 5 personal questions unrelated to mathematics to ask you. Is this ok?

That's fine. I can't guarantee I will have an opinion on every question though. There are some things that I just am not informed enough to have an opinion worth sharing. :)

...I have learned so much from your replies since joining the MHB.

That's always nice to hear! (Yes)

 

[mathdad]

Providing complete step by step instructions for completing a problem generally isn't necessary unless the OP has no idea how to begin the problem...and even then it would really depend on the situation. That could be a great deal of work to do as standard practice. I would rather hear from the OP what they think the steps should be, and then I can offer any advice based on that.
That's fine. I can't guarantee I will have an opinion on every question though. There are some things that I just am not informed enough to have an opinion worth sharing. :)
That's always nice to hear! (Yes)

Although I ask for guidance almost everyday, I know more math than most people in my circle of friends. Most NYC high school graduates do not know the difference between a linear equation and quadratic equation. Most people fear fractions and percents. I feel confident enough to work as a math tutor for grades 1 to 10. This is an accomplishment. One day, I will know precalculus, calculus 1-3 as easily as drinking water. This will take some time but surely doable.

 

[mathdad]

Providing complete step by step instructions for completing a problem generally isn't necessary unless the OP has no idea how to begin the problem...and even then it would really depend on the situation. That could be a great deal of work to do as standard practice. I would rather hear from the OP what they think the steps should be, and then I can offer any advice based on that.
That's fine. I can't guarantee I will have an opinion on every question though. There are some things that I just am not informed enough to have an opinion worth sharing. :)
That's always nice to hear! (Yes)

Check your PM.

 

[mathdad]

I am not going to type the entire calculation. This will take a long time.

1. After completing the square for x^2 - 6x + y^2 - 4y + 12 = 0, I found the center to be (3, 2).

2. After completing the square for x^2 + y^2 - 14x + 47 = 0, I found the center to be (7, 0).

3. I then found the midpoint of the two center points above to be (5, 1).

4. I now need the radius.

r = sqrt{(-4-5)^2 + (1-1)^2}

r = sqrt{(-9)^2 + (0)^2}

r = sqrt{81}

r = 9

5. I now plug the center point (5, 1) and radius 9 into the standard form equation.

(x - 5)^2 + (y - 1)^2 = 81

Is this correct?