Circle Equation Problem
Problem Statement
Find the equation of the circle passing through (4, 1), (6, 5) and having the center on the line 4x + y - 16 = 0.
Solution
Step 1: Understand the Problem
We need to find the equation of a circle that:
- Passes through the points A(4, 1) and B(6, 5)
- Has its center on the line 4x + y - 16 = 0
Step 2: Let the Center be (h, k)
Since the center lies on the line 4x + y - 16 = 0, it satisfies:
4h + k - 16 = 0
⇒ k = 16 - 4h (Equation 1)
Step 3: Use the Distance from Center to Points
The distance from the center (h, k) to A(4, 1) and B(6, 5) must be equal (both equal to the radius r):
√[(h - 4)² + (k - 1)²] = √[(h - 6)² + (k - 5)²]
Square both sides:
(h - 4)² + (k - 1)² = (h - 6)² + (k - 5)²
Expand and simplify:
h² - 8h + 16 + k² - 2k + 1 = h² - 12h + 36 + k² - 10k + 25
-8h - 2k + 17 = -12h - 10k + 61
4h + 8k - 44 = 0
h + 2k - 11 = 0 (Equation 2)
Step 4: Substitute k from Equation 1 into Equation 2
From Equation 1: k = 16 - 4h. Substitute into Equation 2:
h + 2(16 - 4h) - 11 = 0
h + 32 - 8h - 11 = 0
-7h + 21 = 0
h = 3
Now, find k:
k = 16 - 4(3) = 4
So, the center is at (3, 4).
Step 5: Find the Radius
Use the distance from the center to point A(4, 1):
r = √[(3 - 4)² + (4 - 1)²] = √[1 + 9] = √10
Step 6: Write the Equation of the Circle
The standard form of a circle's equation is:
(x - h)² + (y - k)² = r²
Substitute h = 3, k = 4, r² = 10:
(x - 3)² + (y - 4)² = 10
Final Answer
(x - 3)² + (y - 4)² = 10
Circle Equation Problem
Problem Statement
Find the equation of the circle passing through (4, 1), (6, 5) and having the center on the line 4x + y - 16 = 0.
Solution
Final Answer
(x - 3)² + (y - 4)² = 10
Accurate Diagram
Diagram Key:
- Red points: Given points A(4,1) and B(6,5)
- Blue point: Center C(3,4)
- Dashed green line: 4x + y - 16 = 0
- Purple circle: Solution circle with radius √10
Scale: 1 unit = 30 pixels