Math Problem Statement
Find the equation of circle passing through the vertices of triangle whose sides are represented by the equation x+y=2, 3x-4y=6, x-y=0
Solution
To find the equation of the circle passing through the vertices of the triangle formed by the given lines, follow these steps:
Step 1: Find the Vertices of the Triangle
The given lines are:
- (or )
Find the intersection points of these lines:
Vertex A: Intersection of and
Solving simultaneously: Substituting in : So, .
Vertex B: Intersection of and
Substituting in :
Since , we get .
So, .
Vertex C: Intersection of and
Multiply by 3: Now, subtract from : Substituting in : So, .
Step 2: Find the Equation of the Circumcircle
The general equation of a circle is: Substituting the three points , , and into the equation:
Using Point A (1,1):
Using Point B (-6,-6):
Using Point C (2,0):
Step 3: Solve for D, E, and F
We have the system:
Solve for from equation (3):
Substituting in equation (1):
Substituting in equation (2):
Solve equations (4) and (5):
Multiply (4) by 3: Subtract from (5):
Substituting in (4):
Substituting in :
Step 4: Write the Equation of the Circle
Final Answer:
Would you like me to simplify further or explain any step?
Here are 5 related problems you can try:
- Find the circumcenter of the triangle formed by the given three lines.
- Find the radius of the circumcircle of the triangle.
- Find the equation of the centroid of the triangle.
- Find the area of the triangle formed by the given three lines.
- Find the equation of the orthocenter of the triangle.
Tip:
To find the radius of the circumcircle, use the formula: where are the distances between the vertices.
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Circle Equations
Intersection of Lines
Circumcircle
Formulas
General equation of a circle: x^2 + y^2 + Dx + Ey + F = 0
Intersection of two lines (solving system of equations)
Theorems
Circumcenter of a triangle
Equation of Circumcircle
Suitable Grade Level
Grades 11-12
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