a) The equation of a circle with center at (h, k) and radius r is given by (x-h)² + (y-k)² = r². In this case, h = 4, k = 5, and r = 13. So, the equation of the circle is (x-4)² + (y-5)² = 169.

Since points A and B lie on the x-axis, their y-coordinates are 0. Substituting y = 0 into the equation of the circle gives (x-4)² + (0-5)² = 169. Simplifying this gives (x-4)² + 25 = 169, or (x-4)² = 144. Taking the square root of both sides gives x - 4 = ±12. Therefore, the x-coordinates of points A and B are 4 + 12 = 16 and 4 - 12 = -8. So, the coordinates of A and B are (16, 0) and (-8, 0), respectively.

b) The distance between two points (x₁, y₁) and (x₂, y₂) is given by √[(x₂ - x₁)² + (y₂ - y₁)²]. In this case, x₁ = -8, y₁ = 0, x₂ = 16, and y₂ = 0. So, the distance between A and B is √[(16 - (-8))² + (0 - 0)²] = √[24²] = 24 units.

Question

a) The equation of a circle with center at (h, k) and radius r is given by (x-h)² + (y-k)² = r². In this case, h = 4, k = 5, and r = 13. So, the equation of the circle is (x-4)² + (y-5)² = 169.

Since points A and B lie on the x-axis, their y-coordinates are 0. Substituting y = 0 into the equation of the circle gives (x-4)² + (0-5)² = 169. Simplifying this gives (x-4)² + 25 = 169, or (x-4)² = 144. Taking the square root of both sides gives x - 4 = ±12. Therefore, the x-coordinates of points A and B are 4 + 12 = 16 and 4 - 12 = -8. So, the coordinates of A and B are (16, 0) and (-8, 0), respectively.

b) The distance between two points (x₁, y₁) and (x₂, y₂) is given by √[(x₂ - x₁)² + (y₂ - y₁)²]. In this case, x₁ = -8, y₁ = 0, x₂ = 16, and y₂ = 0. So, the distance between A and B is √[(16 - (-8))² + (0 - 0)²] = √[24²] = 24 units.

Knowee AI · Accepted Answer

a) The equation of a circle with center at (h, k) and radius r is given by (x-h)² + (y-k)² = r². In this case, h = 4, k = 5, and r = 13. So, the equation of the circle is (x-4)² + (y-5)² = 169.

Since points A and B lie on the x-axis, their y-coordinates are 0. Substituting y = 0 into the equation of the circle gives (x-4)² + (0-5)² = 169. Simplifying this gives (x-4)² + 25 = 169, or (x-4)² = 144. Taking the square root of both sides gives x - 4 = ±12. Therefore, the x-coordinates of points A and B are 4 + 12 = 16 and 4 - 12 = -8. So, the coordinates of A and B are (16, 0) and (-8, 0), respectively.

b) The distance between two points (x₁, y₁) and (x₂, y₂) is given by √[(x₂ - x₁)² + (y₂ - y₁)²]. In this case, x₁ = -8, y₁ = 0, x₂ = 16, and y₂ = 0. So, the distance between A and B is √[(16 - (-8))² + (0 - 0)²] = √[24²] = 24 units.

When a circle of radius 13 unit and mid-point (4, 5) is drawn, the point A and B cuts the x-axis.a) Find the coordinates of A and Bb) Find the length of AB.

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