Need math help Tangent to circle question

[BCfortheWin]

The question is:

A circle touches the y-axis at the origin and goes through the point A(8, 0). The point C is
on the circumference. Find the greatest possible area of ∆OAC

I graphed the above situation, and used the equation A=(1/2)bcsinA, but i couldn't muster up an answer.

Your help would be much appreciated. Thank you!

 

[CompuChip]

Just to check that you did it correctly, can you describe the circle (e.g. what is its center and radius)?

For simplicity, let's just assume that C is in the top half of the plane, so y > 0.

Two questions:
1) Suppose it has coordinates (x, y) - then what is the area A?
2) What is the relation between x and y?

 

[BCfortheWin]

Im not quite sute what you mean...i conveniently made the center (4,0), can i do that? Hence the triangle is in the upper half of the semi-circle! I conveniently (again!) put C at the middle of the semi-cirlce's circumference, hence making the triangle an isosceles right triangle.

With the above set-up i was able to get the answer, but I am not quite sure why the setup had to be above. Can you solve the question and tell me what you got, and how and why solved it that way? Thanks alot!

 

[putongren]

Hey, this is my take on the problem.

I would put the coordinates of the center of the circle (denoted as J) as (4, y) and C as (4, 2y) (Do you know why?). We need to know the value of y. Let angle JOA = θ and JOC = θ , and let the line segment JO and line segment AO equal the radius of the circle.

sin θ = ^y/_r, cos θ = ^r/4, solve y in terms of sin θ and cos θ, Once you get y, you get C, which is the height of the ΔOAC. Since you know the base of the triangle, you should get an expression of the area in terms of θ. Differentiate the area in terms of θ to find the max area. (Are you allowed to use calculus?)

 

[BCfortheWin]

Hey, this is my take on the problem.

I would put the coordinates of the center of the circle (denoted as J) as (4, y) and C as (4, 2y) (Do you know why?). We need to know the value of y. Let angle JOA = θ and JOC = θ , and let the line segment JO and line segment AO equal the radius of the circle.

sin θ = ^y/_r, cos θ = ^r/4, solve y in terms of sin θ and cos θ, Once you get y, you get C, which is the height of the ΔOAC. Since you know the base of the triangle, you should get an expression of the area in terms of θ. Differentiate the area in terms of θ to find the max area. (Are you allowed to use calculus?)

Your not allowed to used calculus

 

[zgozvrm]

Technically, the answer is "infinite" since the question doesn't state that the circle only touches the y-axis at the origin (in other words, it doesn't state that the circle is tangent to the y-axis). Therefore, you are given only 2 points through which the circle passes and there are an infinite number of circles that pass through those 2 points.


Otherwise (if the circle is indeed tangent to the y-axis at the origin), then the question could be more simply stated as:

Given a circle with a radius of 4 units, let points A and B be the endpoints of a diameter of the circle. Let point C be any other point on that circle. Find the maximum area of triangle ABC.

 

[CompuChip]

Hi, sorry for the delayed reply.

Here is my take on the problem. First of all, it is not just convenient to set the center to (4, 0) - it is necessary. Try drawing some circles on a piece of paper... the only way to get it to be tangent to the y-axis in the origin (you said "touch" but the topic title mentions "tangent" - so I assume that is what you meant) is if the center lies on the x-axis. Seriously: work it out and see for yourself. So then it is easy to see where the center should be: if the circle should cross the x-axis at x = 0 and at x = 8, the center should be at x = (8 + 0) / 2 = 4.

Now, consider an arbitrary point on the upper semi-circle, and let's call its coordinates (x, y). Then if you draw an image, you can express the area of the triangle A in terms of x and y - or, as it so happens, in terms of y only in this case. When you do that, you should get A = 4y. It doesn't take calculus to see that A has a maximum wherever y has a maximum.