Find The Equation Of The Circle That Passes Through The Points { (2,3)$}$ And { (-1,2)$}$, And Has Its Center On The Line ${ 2x - 3y + 1 = 0\$} .
Introduction
In mathematics, the equation of a circle is a fundamental concept that is used to describe the shape and position of a circle in a two-dimensional plane. The general equation of a circle is given by , where is the center of the circle and is the radius. In this article, we will discuss how to find the equation of a circle that passes through two given points and has its center on a given line.
Given Information
We are given two points and through which the circle passes, and a line on which the center of the circle lies.
Step 1: Find the Equation of the Line
The given line is . We can rewrite this equation in the slope-intercept form , where is the slope and is the y-intercept.
import sympy as sp

x, y = sp.symbols('x y')
line_eq = 2x - 3y + 1
y_sol = sp.solve(line_eq, y)[0]
print(y_sol)
The equation of the line in slope-intercept form is .
Step 2: Find the Slope of the Line
The slope of the line is the coefficient of in the slope-intercept form. In this case, the slope is .
Step 3: Find the Equation of the Perpendicular Bisector
The perpendicular bisector of the line segment joining the two points is the line that passes through the midpoint of the line segment and is perpendicular to the line segment. The equation of the perpendicular bisector is given by , where is the midpoint of the line segment and is the slope of the perpendicular bisector.
# Define the coordinates of the two points
x1, y1 = 2, 3
x2, y2 = -1, 2
midpoint_x = (x1 + x2) / 2
midpoint_y = (y1 + y2) / 2
m = -1 / (2/3)
perp_bisector_eq = y - midpoint_y - m * (x - midpoint_x)
print(perp_bisector_eq)
The equation of the perpendicular bisector is .
Step 4: Find the Intersection of the Perpendicular Bisector and the Given Line
The intersection of the perpendicular bisector and the given line is the point where the two lines intersect. We can find the intersection by solving the two equations simultaneously.
# Define the equation of the given line
given_line_eq = 2*x - 3*y + 1
intersection = sp.solve((perp_bisector_eq, given_line_eq), (x, y))
print(intersection)
The intersection of the perpendicular bisector and the given line is .
Step 5: Find the Equation of the Circle
The equation of the circle is given by , where is the center of the circle and is the radius. We can find the equation of the circle by substituting the coordinates of the center and the radius into the general equation.
# Define the coordinates of the center
h, k = -1/2, 5/2
r = sp.sqrt((2 - h)**2 + (3 - k)**2)
circle_eq = (x - h)**2 + (y - k)2 - r2
print(circle_eq)
The equation of the circle is .
Conclusion
In this article, we discussed how to find the equation of a circle that passes through two given points and has its center on a given line. We used the concept of the perpendicular bisector and the intersection of two lines to find the center of the circle. We then used the general equation of a circle to find the equation of the circle. The final equation of the circle is .
Q: What is the general equation of a circle?
A: The general equation of a circle is given by , where is the center of the circle and is the radius.
Q: How do I find the equation of a circle that passes through two given points and has its center on a given line?
A: To find the equation of a circle that passes through two given points and has its center on a given line, you need to follow these steps:
- Find the equation of the perpendicular bisector of the line segment joining the two points.
- Find the intersection of the perpendicular bisector and the given line.
- Use the coordinates of the intersection point as the center of the circle.
- Use the distance formula to find the radius of the circle.
- Substitute the coordinates of the center and the radius into the general equation of a circle.
Q: What is the perpendicular bisector of a line segment?
A: The perpendicular bisector of a line segment is a line that passes through the midpoint of the line segment and is perpendicular to the line segment.
Q: How do I find the equation of the perpendicular bisector of a line segment?
A: To find the equation of the perpendicular bisector of a line segment, you need to follow these steps:
- Find the midpoint of the line segment.
- Find the slope of the line segment.
- Find the slope of the perpendicular bisector by taking the negative reciprocal of the slope of the line segment.
- Use the point-slope form of a line to find the equation of the perpendicular bisector.
Q: What is the intersection of two lines?
A: The intersection of two lines is the point where the two lines meet.
Q: How do I find the intersection of two lines?
A: To find the intersection of two lines, you need to solve the two equations simultaneously.
Q: What is the distance formula?
A: The distance formula is a formula used to find the distance between two points in a coordinate plane.
Q: How do I use the distance formula to find the radius of a circle?
A: To use the distance formula to find the radius of a circle, you need to follow these steps:
- Find the coordinates of the center of the circle.
- Find the coordinates of one of the points on the circle.
- Use the distance formula to find the distance between the center and the point.
- This distance is the radius of the circle.
Q: What is the equation of a circle in standard form?
A: The equation of a circle in standard form is given by , where is the center of the circle and is the radius.
Q: How do I convert the equation of a circle from general form to standard form?
A: To convert the equation of a circle from general form to standard form, you need to follow these steps:
- Complete the square for the x-terms.
- Complete the square for the y-terms.
- Simplify the equation to get the standard form.
Q: What is the significance of the center and radius of a circle?
A: The center and radius of a circle are important because they determine the shape and size of the circle.
Q: How do I find the equation of a circle that passes through a given point and has its center on a given line?
A: To find the equation of a circle that passes through a given point and has its center on a given line, you need to follow these steps:
- Find the equation of the perpendicular bisector of the line segment joining the given point and the center of the circle.
- Find the intersection of the perpendicular bisector and the given line.
- Use the coordinates of the intersection point as the center of the circle.
- Use the distance formula to find the radius of the circle.
- Substitute the coordinates of the center and the radius into the general equation of a circle.
Q: What is the equation of a circle in parametric form?
A: The equation of a circle in parametric form is given by and , where is the center of the circle and is the radius.
Q: How do I convert the equation of a circle from general form to parametric form?
A: To convert the equation of a circle from general form to parametric form, you need to follow these steps:
- Use the trigonometric identities and to rewrite the equation in parametric form.
Q: What is the significance of the parametric form of a circle?
A: The parametric form of a circle is important because it allows us to describe the circle in terms of a parameter, which can be useful in certain applications.
Q: How do I find the equation of a circle that passes through a given point and has its center on a given line in parametric form?
A: To find the equation of a circle that passes through a given point and has its center on a given line in parametric form, you need to follow these steps:
- Find the equation of the perpendicular bisector of the line segment joining the given point and the center of the circle.
- Find the intersection of the perpendicular bisector and the given line.
- Use the coordinates of the intersection point as the center of the circle.
- Use the distance formula to find the radius of the circle.
- Substitute the coordinates of the center and the radius into the parametric form of a circle.
Conclusion
In this article, we have discussed some frequently asked questions about finding the equation of a circle. We have covered topics such as the general equation of a circle, the perpendicular bisector, the intersection of two lines, the distance formula, and the parametric form of a circle. We have also provided examples and step-by-step instructions on how to find the equation of a circle in different forms.