Finding the equation of the circle given two points and a tangent line

In summary, to find the equation of the circle that passes through the point (3,-2) and is tangent to the line y=3x+5 at (-1,2), you need to find the center (h,k) and the radius r. If you know calculus, you can use the fact that the two given points must satisfy the equation of the circle and the slope of the circle must equal the slope of the tangent line. If you cannot use calculus, you can find the center by finding the point on the line perpendicular to the tangent line at (-1,2) that is equidistant from the two given points. Once you have the center, you can use the standard form equation (x-h)^2+(
  • #1
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Homework Statement


Find the equation of the circle that passes through the point (3,-2) and tangent to the line y=3x+5 at (-1,2). Answer in standard form.


Homework Equations


d= |mx0+b-y0|
____________
sqrt(1+m2

is needed to find the radius of the circle
(x-h)2+(y-k)2=r2 is needed to find the equation of the circle.

The Attempt at a Solution



I haven't attempted a solution per se, but I do know that I need to find the center of the circle. However I don't see a way to find the center of a circle.
 
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  • #2
You need to figure out h, k, and r, right? Three unknowns. Here are a couple of suggestions, depending on whether you know any calculus.

If you know calculus, you can get three equations in the three unknowns by using the fact that your two given points must work in the equation, and the slope of the circle must equal the slope of the tangent line where they touch.

If you can't use calculus, write the equation of the line perpendicular to the tangent line at (-1,2) and find the point on that line that is equidistant from your two given points. That will be your center.
 

1. How do you find the equation of a circle given two points and a tangent line?

To find the equation of a circle given two points and a tangent line, you will need to use the formula (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle and r is the radius. The two given points will serve as the endpoints of the diameter of the circle. You will also need to use the slope of the tangent line to determine the radius of the circle.

2. What information do I need to know in order to find the equation of a circle given two points and a tangent line?

In order to find the equation of a circle given two points and a tangent line, you will need to know the coordinates of the two points, as well as the slope of the tangent line. This information will allow you to determine the center and radius of the circle, which are necessary for the equation.

3. Can I use the distance formula to find the equation of a circle given two points and a tangent line?

Yes, you can use the distance formula, which is d = √[(x1-x2)^2 + (y1-y2)^2], to find the radius of the circle. You can use the distance between the two given points as the diameter of the circle, and then divide it by 2 to find the radius.

4. Is it possible to have more than one circle that satisfies the given conditions of two points and a tangent line?

Yes, it is possible to have more than one circle that satisfies the given conditions. This is because there are an infinite number of circles that can pass through two given points and have a tangent line at a certain point. However, each circle will have a different center and radius, resulting in a different equation.

5. What if the given points are collinear? Can I still find the equation of a circle?

If the given points are collinear, meaning they are all on the same line, then it is not possible to find the equation of a circle that passes through those points. This is because a circle must have a center and a radius, which cannot be determined if all the points are on the same line.

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