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The equation of circle passing through (4,5) and having centre (2,2)

The equation of circle passing through (4,5) and having centre (2,2)

Grade:11

1 Answers

Arun
25757 Points
5 years ago
1. (4, 5) lies on the circle. Use the distance formula to find the distance from the center, (2, 2) to (4, 5). That would be the radius. 

d = √[ (x₂ - x₁)² + (y₂ - y₁)² ] 
r  = √[ (4 - 2)² + (5 - 2)² ] 
r  = √13 

The equation of a circle in standard form is 
(x - a)² + (y - b)² = r²          center = (a, b) 

(x - 2)² + (y - 2)² = (√13)² 
(x - 2)² + (y - 2)² = 13 

2. You need to find the equation of a circle with the same center as x² + y² - 4x - 6y + 9 = 0. 

x² + y² - 4x - 6y + 9 = 0 
(x² - 4x) + (y² - 6y) = -9 

To complete the square, take half of 4 and half of 6, then square it and add it to both sides. 
(4/2)² = 4 
(6/2)² = 9 

(x² - 4x + 4) + (y² - 6y + 9) = -9 + 4 + 9 
(x - 2)(x - 2) + (y - 3)(y - 3) = 4 
(x - 2)² + (y - 3)² = 2² 

center = (2, 3) 

To be concentric, the circle we're looking for also has a center at (2, 3). Since (-4, 5) lies on this circle, we can find the radius by finding the distance between those points. 

r = √[ (-4 - 2)² + (5 - 3)² ] 
r = √40 

(x - a)² + (y - b)² = r²          center = (a, b) 
(x - 2)² + (y - 3)² = (√40)² 
(x - 2)² + (y - 3)² = 40 

3. 
(1, -2) lies on the circle. 
(1 - a)² + (-2 - b)² = r² 

(4, -3) also lies on the circle. 
(4 - a)² + (-3 - b)² = r² 

Since both equations have the same radius, they are equal to each other. 
(1 - a)² + (-2 - b)² = (4 - a)² + (-3 - b)² 
1 - 2a + a² + 4 + 4b + b² = 16 - 8a + a² + 9 + 6b + b² 
a² - a² + b² - b² - 2a + 8a + 4b - 6b + 1 + 4 - 9 - 16 = 0 
6a - 2b - 20 = 0 

(a, b) is the center. Since the center lies on the line 3x + 4y = 7, the following holds true: 
3a + 4b = 7 

6a - 2b - 20 = 0 
6a - 20 = 2b 
(6a - 20) / 2 = b 
3a - 10 = b 

3a + 4b = 7 
3a + 4(3a - 10) = 7 
3a + 12a - 40 = 7 
15a = 47 
a = 47/15 
a = 3.133 

3a - 10 = b 
3(47/15) - 10 = b 
-3/5 = b 
-0.6 = b 

center = (a, b) = (3.133, -0.6) 

Pick one of the points to find the radius. I'll use (1, -2). 
r = √[ (1 - 3.133)² + (-2 - (-0.6))² ] 
r = √6.51 

Answer: 
(x - a)² + (y - b)² = r² 
(x - 3.133)² + (y - (-0.6))² = (√6.51)² 
(x - 3.133)² + (y + 0.6)² = 6.51 

Check: 
(1,-2) and (4,-3) lie on the circle, so plug them in to check. 
(1 - 3.133)² + (-2 + 0.6)² = 6.51 
(4 - 3.133)² + (-3 + 0.6)² = 6.51 

The center coordinates lie on the line, so 
3(3.133) + 4(-0.6) = 7

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