# Need math help! Tangent to circle question

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 couldnt muster up an answer.

Your help would be much appreciated. Thank you!

Last edited:

CompuChip
Homework Helper
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?

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 im 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!

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?)

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

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