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

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3 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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3 years ago
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