Find the equation of a circle passing through (3,7) and tangent to the line x-3y+8=0.
Please, How do I solve this? Help me
To find the equation of a circle passing through a given point and tangent to a given line, you'll need to follow a few steps:
Step 1: Find the center of the circle.
- Since the circle passes through (3,7), the center of the circle will lie on the perpendicular bisector of the line connecting (3,7) to the point of tangency.
- Find the slope of the line: The given line is x - 3y + 8 = 0. Rearrange it to slope-intercept form, which is y = (1/3)x + 8/3. The slope of this line, m, is 1/3.
- The slope of the perpendicular bisector will be the negative reciprocal of m. So, the slope of the perpendicular bisector is -3.
- Now, you have the midpoint between (3,7) and the point of tangency. Use the midpoint formula to find the coordinates of the center of the circle.
Step 2: Find the radius of the circle.
- The radius of the circle is the distance between the center and any point on the circle.
- You already have the coordinates of the center and a point on the circle (3,7). Use the distance formula to find the radius.
Step 3: Write the equation of the circle.
- Once you have the center coordinates and the radius, you can write the equation of the circle using the formula (x - h)^2 + (y - k)^2 = r^2, where (h, k) are the coordinates of the center and r is the radius.
Now, let's solve this problem step by step.
Step 1: Find the center of the circle.
- The slope of the given line is 1/3. The slope of the perpendicular bisector will be -3.
- The midpoint formula is: (x1 + x2)/2, (y1 + y2)/2.
- Using the midpoint formula: ((3 + x)/2, (7 + y)/2) = (-3, 0).
- Solve for x and y to find the coordinates of the center.
Step 2: Find the radius of the circle.
- Use the distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2).
- Substitute the coordinates of the center (x, y) and the point on the circle (3,7) into the formula.
- Solve for the radius.
Step 3: Write the equation of the circle.
- Now that you have the center coordinates (h, k) and the radius r, you can write the equation of the circle as (x - h)^2 + (y - k)^2 = r^2.
By following these steps, you can solve the problem and find the equation of the circle passing through (3,7) and tangent to the line x-3y+8=0.