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Grade: 11

                        

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

3 years ago

Answers : (1)

Arun
24736 Points
							
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
3 years ago
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