Finding equation of a circle given circumference and containing points.

[anniecvc]

Homework Statement

Find the equation of a circle if the circumference is 18∏ and contains the point (2, 8)

The Attempt at a Solution

I know I can find the radius by setting 18∏=2∏r. r=9.

the equation of a circle is (x-h)²+(y-k)²=r²

So I have 9²= (2-h)²+(8-k)²

which becomes 81= 4-4h+h²+64-16k+k²

I can simplify this down to -4h + h² +68 -16k +k² = 81


To my understanding, containing the point does not mean the center, but on the circumference of the circle. I'm lost.

 

[d2j2003]

if you know that the radius is 9 you can figure out the equation of the circle centered at the origin and then just translate it so that your point lies a distance of r from the center.

 

[PeterO]

Homework Statement

Find the equation of a circle if the circumference is 18∏ and contains the point (2, 8)

The Attempt at a Solution

I know I can find the radius by setting 18∏=2∏r. r=9.

the equation of a circle is (x-h)²+(y-k)²=r²

So I have 9²= (2-h)²+(8-k)²

which becomes 81= 4-4h+h²+64-16k+k²

I can simplify this down to -4h + h² +68 -16k +k² = 81


To my understanding, containing the point does not mean the center, but on the circumference of the circle. I'm lost.

I don't believe there is enough information.

If you were to model this by drawing axes, and using a CD to represent the circle, you are only told that the point 2,8 is one of the points on the circle.
It is possible to place one point on the circumference of the CD at the point (2,8), but then rotate the CD slowly, keeping the given point of the CD at (2,8).

I believe you would need one other reference point in order to define a single location for the model - and thus the answer you seek.

 

[PeterO]

if you know that the radius is 9 you can figure out the equation of the circle centered at the origin and then just translate it so that your point lies a distance of r from the center.

But the translation could be in the +y direction, the -y direction, the +x direction, the -x direction or in an infinite number of oblique directions.

 

[d2j2003]

right, but it would give an equation for a circle containing that point.. just not a specific equation because, as you said, you would need another point for that

 

[PeterO]

right, but it would give an equation for a circle containing that point.. just not a specific equation because, as you said, you would need another point for that

Unfortunately the original task was to:

Find the equation of a circle if the circumference is 18∏ and contains the point (2, 8)

not

Find an equation of a circle if the circumference is 18∏ and contains the point (2, 8)

 

[NascentOxygen]

Unfortunately the original task was to:

Find the equation of a circle if the circumference is 18∏ and contains the point (2, 8)

not

Find an equation of a circle if the circumference is 18∏ and contains the point (2, 8)

I would say that the question can be restated equivalently as: Find the equation of any circle of circumference 18π and containing the point (2,8)." The telling point is that it does not say "of the circle".

This appears to be a case where, were it an exam question, in answering it you can "only do your best". The examiner may not realize his mistake (if indeed one exists) until he begins to mark the examinees' returned papers. Besides, the careless grammar in the question's wording can give little confidence that it accurately reflects what the examiner may have had in mind.

 

[HallsofIvy]

There are an infinite number of circles of a given radius containing a given point. Taking the given point as center, construct a circle of the given radius. You can take any point on that circle as center of your desired circle.