Finding the equation of a certain circle

[Dev2011]

My Honors Algebra II class has been assigned some extra credit problems that seem to be QUITE difficult. I desperately need these points, so any help would be appreciated. Thanks so much!

Here is one of our problems:

Find the equation of the circle centered at the origin and tangent to the line 3x+4y+50. Use the following steps:
A) FInd the slope of the given tangent line, and use it to write an equation for the line containing the origin and point of tangency. (HInt- Call the point of tangency (x,y).)
B) Solve the system consisting of the equation you wrote in part A and the given equation to find the coordinates of the point of tangency. Use these to find the radius of the circle, and its equation.

 

[Mrspi]

My Honors Algebra II class has been assigned some extra credit problems that seem to be QUITE difficult. I desperately need these points, so any help would be appreciated. Thanks so much!

Here is one of our problems:

Find the equation of the circle centered at the origin and tangent to the line 3x+4y+50. Use the following steps:
A) FInd the slope of the given tangent line, and use it to write an equation for the line containing the origin and point of tangency. (HInt- Call the point of tangency (x,y).)
B) Solve the system consisting of the equation you wrote in part A and the given equation to find the coordinates of the point of tangency. Use these to find the radius of the circle, and its equation.

So...what have you done to apply the given hints?

You are the one who desperately needs these points, so I surely expect to see some effort on your part before I "jump in."

 

[Dev2011]

I know that I first need to find the slope of 3x+4y=50. I believe that that is -3/4. From there, I need to find the line that contains (0,0) and the point of tangency. I have never worked with points of tangency before, so I am not sure how to find where on the circle that is. That is my first hurdle.

 

[Mrspi]

I know that I first need to find the slope of 3x+4y=50. I believe that that is -3/4. From there, I need to find the line that contains (0,0) and the point of tangency. I have never worked with points of tangency before, so I am not sure how to find where on the circle that is. That is my first hurdle.

You are correct...the slope of the line whose equation is 3x + 4y = 50 is -3/4.

Now, you are expected to know that a radius drawn to the point of tangency is PERPENDICULAR to the tangent line (which is 3x + 4y = 50)

What do you know about the slopes of perpendicular lines?

 

[Dev2011]

I'm not sure how to say this mathematically, but the slopes of perpendicular lines are opposite and flipped. so in this case, the slope of the line that the point of tangency is on is 4/3. So now we know at least 2 points that are on this line: (0,0) and (4,3). With that, i think we can find the equation of the line. y=3/4x?

 

[Mrspi]

I'm not sure how to say this mathematically, but the slopes of perpendicular lines are opposite and flipped. so in this case, the slope of the line that the point of tangency is on is 4/3. So now we know at least 2 points that are on this line: (0,0) and (4,3). With that, i think we can find the equation of the line. y=3/4x?

Not exactly.

The slope of the perpendicular line is indeed 4/3.

You know the line from the center through the point of tangency has a slope of 4/3, and contains the point (0, 0)

your equation SHOULD be y = (4/3)x

Now, the point of tangency is the intersection of the tangent line,

3x + 4y = 50

and the radius to the point of tangency, which is part of the line

y = (4/3)x

Solve this system of equations to find the coordinates of the point ON the circle where the radius intersects the tangent line (this is the point of tangency).

The distance from the center point (0, 0) to the point of tangency is the radius of the circle.

The standard form for the equation of a circle with center at (h, k) and radius "r" is

(x - h)[sup:30w9lmqr]2[/sup:30w9lmqr] + (y - k)[sup:30w9lmqr]2[/sup:30w9lmqr] = r[sup:30w9lmqr]2[/sup:30w9lmqr]

Since this is an extra credit problem, I'll expect YOU to finish it.

 

[Dev2011]

Okay, I think I've got it.

I found that the point of tangency is (7.5, 10)

that the radius is 12.5

and that the equation of the circle is x[sup:1msrrq5o]2[/sup:1msrrq5o]+ y[sup:1msrrq5o]2[/sup:1msrrq5o]= 12.5[sup:1msrrq5o]2[/sup:1msrrq5o]

am i correct?

 

[Dev2011]

I re-worked my math and changed my mind.

the point of tangency is (6,8)

the radius is 10

and the equation of the circle is x[sup:3dy74w94]2[/sup:3dy74w94]+y[sup:3dy74w94]2[/sup:3dy74w94]=100

how about that?

 

[Mrspi]

I re-worked my math and changed my mind.

the point of tangency is (6,8)

the radius is 10

and the equation of the circle is x[sup:1xumqyus]2[/sup:1xumqyus]+y[sup:1xumqyus]2[/sup:1xumqyus]=100

how about that?

Well, you can (and SHOULD) check your solution.

You have the point of tangency as (6, 8)....

See if this point satisfies the given equation 3x + 4y = 50 (if it does, you know that the point ON the circle is the intersection of the tangent line and the circle).