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
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To find the equation of a circle passing through a given point and tangent to a given line, you can follow these steps:
Step 1: Find the center of the circle:
- A circle passing through a given point will have its center somewhere on the perpendicular bisector of the line joining the given point and the center of the circle.
- Find the midpoint of the line segment joining the given point and the center of the circle.
- The coordinates of this midpoint will give you the center of the circle.
Step 2: Find the radius of the circle:
- The distance between the center of the circle and the given point is equal to the radius of the circle.
- Use the distance formula to find the distance between the center and the given point.
Step 3: Write the equation of the circle:
- Use the center and radius obtained in the previous steps to write the equation of the circle in the general form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
Now, let's solve the problem using these steps:
Step 1: Finding the center of the circle:
- The equation of the given line is x - 3y + 8 = 0.
- Rewrite this equation in slope-intercept form: y = (1/3)x + 8/3.
- The slope of this line is 1/3, so the slope of the perpendicular bisector will be -3 (negative reciprocal).
- Using the midpoint formula: (x, y) = ((3 + h)/2, (7 + k)/2).
Step 2: Finding the radius of the circle:
- The distance between the center of the circle and the given point (3, 7) is equal to the radius.
- Use the distance formula: √((x2 - x1)^2 + (y2 - y1)^2).
Step 3: Writing the equation of the circle:
- Substitute the center and radius found in the previous steps into the general equation form: (x - h)^2 + (y - k)^2 = r^2.
Now, let's solve the problem step by step.
Step 1: Finding the center of the circle:
- Using the slope-intercept form of the given line, y = (1/3)x + 8/3, we can conclude that the slope of the perpendicular bisector is -3.
- The midpoint formula gives us the center of the circle, (h, k), as ((3 + h)/2, (7 + k)/2).
Step 2: Finding the radius of the circle:
- The distance between the center of the circle and the given point (3, 7) is equal to the radius.
- Using the distance formula, which is √((x2 - x1)^2 + (y2 - y1)^2), we can find the radius.
Step 3: Writing the equation of the circle:
- Substitute the values of the center and radius into the general equation (x - h)^2 + (y - k)^2 = r^2 to get the equation of the circle.
Following these steps, you can find the equation of the circle passing through (3,7) and tangent to the line x - 3y + 8 = 0.