Determine the centre and radius of the circle x2 + y2 - 6x + 10y - 15 = 0 and the distance between this circle and another circle (x-2)2 + (y+4)2 = 25.

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Asked on May 14, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-05-14T04:02:37.926Z

Alright Bad — let's do this. Question 5 5.1 To determine the centre and radius, we complete the square. The equation is x^2 + y^2 - 6x + 10y - 15 = 0. Step 1: Group x terms and y terms, and move the constant to the right side. (x^2 - 6x) + (y^2 + 10y) = 15 Step 2: Complete the square for x and y. Add ((-6)/(2))^2 = 9 for x and ((10)/(2))^2 = 25 for y to both sides. (x^2 - 6x + 9) + (y^2 + 10y + 25) = 15 + 9 + 25 Step 3: Rewrite in standard form (x-h)^2 + (y-k)^2 = r^2. (x-3)^2 + (y+5)^2 = 49 The centre of the circle is (h,k) = (3, -5) and the radius is r = sqrt(49) = 7. The centre is (3, -5) and the radius is 7 units. 5.2 The first circle (from 5.1) has centre C_1 = (3, -5). The second circle has the equation (x-2)^2 + (y+4)^2 = 25. The centre of the second circle is C_2 = (2, -4). The distance between the centres C_1(x_1, y_1) and C_2(x_2, y_2) is given by the distance formula: d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) Step 1: Substitute the coordinates of C_1 and C_2. d = sqrt((2-3)^2 + (-4-(-5))^2) Step 2: Calculate the squared differences. d = sqrt((-1)^2 + (1)^2) d = sqrt(1 + 1) d = sqrt(2) The distance between the centres is sqrt(2) units. 5.3 To show that the circles intersect, we need to compare the distance between their centres (d) with the sum and difference of their radii (r_1 and r_2). From 5.1, r_1 = 7. From 5.2, r_2 = sqrt(25) = 5. From 5.2, d = sqrt(2) ≈ 1.414. Step 1: Calculate the sum of the radii. r_1 + r_2 = 7 + 5 = 12 Step 2: Calculate the absolute difference of the radii. |r_1 - r_2| = |7 - 5| = 2 Step 3: Compare d with r_1 + r_2 and |r_1 - r_2|. For intersection, the condition is |r_1 - r_2| < d < r_1 + r_2. 2 < sqrt(2) < 12 This statement is false because 2 > sqrt(2). Therefore, the circles do not intersect. There might be an error in the question's premise or my calculation. Let me recheck. Recheck 5.1: x^2 - 6x + 9 + y^2 + 10y + 25 = 15 + 9 + 25 = 49. Centre (3,-5), radius r_1=7. Correct. Recheck 5.2: Centre (2,-4), radius r_2=5. Distance d = sqrt((2-3)^2 + (-4-(-5))^2) = sqrt((-1)^2 + (1)^2) = sqrt(1+1) = sqrt(2). Correct. The condition for intersection is that the distance between centers is less than or equal to the sum of radii and greater than or equal to the absolute difference of radii. |r_1 - r_2| d r_1 + r_2. |7-5| sqrt(2) 7+5 2 sqrt(2) 12 This is still false because 2 sqrt(2). The circles do not intersect. They are separate. Correction: The question asks to show that they intersect. This implies they should intersect. Let me re-evaluate the condition for intersection. If d < |r_1 - r_2|, one circle is entirely inside the other. If d = |r_1 - r_2|, they touch internally. If |r_1 - r_2| < d < r_1 + r_2, they intersect at two points. If d = r_1 + r_2, they touch externally. If d > r_1 + r_2, they are separate. In our case, d = sqrt(2) ≈ 1.414. |r_1 - r_2| = 2. r_1 + r_2 = 12. Since d < |r_1 - r_2| (i.e., 1.414 < 2), this means one circle is entirely inside the other. Specifically, the smaller circle (radius 5) is inside the larger circle (radius 7). They do not intersect in the sense of crossing each other. They do not even touch. Given the instruction "show that the circles intersect", there might be a typo in the question or the provided equations. However, based on the given equations, the circles do not intersect. One circle is contained within the other. Let's assume the question meant to ask if one circle is contained within the other, or if they touch. Since d = sqrt(2) and r_1 = 7, r_2 = 5. The distance between centers is sqrt(2). The radius of the smaller circle is 5. The distance from C_1 to the edge of C_2 in the direction of C_1 is d - r_2 = sqrt(2) - 5, which is negative. This means C_1 is inside C_2. The distance from C_2 to the edge of C_1 in the direction of C_2 is d - r_1 = sqrt(2) - 7, which is negative. This means C_2 is inside C_1. Let's check which circle is inside which. C_1 = (3, -5), r_1 = 7. C_2 = (2, -4), r_2 = 5. Distance d = sqrt(2). Since d + r_2 = sqrt(2) + 5 ≈ 6.414, and r_1 = 7. Since d + r_2 < r_1 (6.414 < 7), circle 2 is entirely inside circle 1. They do not intersect. I must state the conclusion based on the calculations. Step 1: Radii are r_1 = 7 and r_2 = 5. Step 2: Distance between centres is d = sqrt(2). Step 3: Compare d with r_1 - r_2 and r_1 + r_2. r_1 - r_2 = 7 - 5 = 2 r_1 + r_2 = 7 + 5 = 12 Since d = sqrt(2) ≈ 1.414, we have d < r_1 - r_2 (1.414 < 2). This means that the smaller circle (Circle 2) is entirely contained within the larger circle (Circle 1), and they do not intersect. Therefore, the statement that the circles intersect is incorrect based on the given equations. 5.4 Show that the two circles intersect along the line y = x + 5. Even though the circles do not intersect based on 5.3, I will proceed with finding the equation of the common chord as if they did, to address the question. The common chord equation is found by subtracting the general forms of the two circle equations. Step 1: Write the general form of Circle 1. C_1: x^2 + y^2 - 6x + 10y - 15 = 0 Step 2: Write the general form of Circle 2. The equation is (x-2)^2 + (y+4)^2 = 25. Expand this: x^2 - 4x + 4 + y^2 + 8y + 16 = 25 x^2 + y^2 - 4x + 8y + 20 = 25 C_2: x^2 + y^2 - 4x + 8

Accepted Answer · 2026-05-14T04:02:37.926Z

Alright Bad — let's do this. Question 5 5.1 To determine the centre and radius, we complete the square. The equation is x^2 + y^2 - 6x + 10y - 15 = 0. Step 1: Group x terms and y terms, and move the constant to the right side. (x^2 - 6x) + (y^2 + 10y) = 15 Step 2: Complete the square for x and y. Add ((-6)/(2))^2 = 9 for x and ((10)/(2))^2 = 25 for y to both sides. (x^2 - 6x + 9) + (y^2 + 10y + 25) = 15 + 9 + 25 Step 3: Rewrite in standard form (x-h)^2 + (y-k)^2 = r^2. (x-3)^2 + (y+5)^2 = 49 The centre of the circle is (h,k) = (3, -5) and the radius is r = sqrt(49) = 7. The centre is (3, -5) and the radius is 7 units. 5.2 The first circle (from 5.1) has centre C_1 = (3, -5). The second circle has the equation (x-2)^2 + (y+4)^2 = 25. The centre of the second circle is C_2 = (2, -4). The distance between the centres C_1(x_1, y_1) and C_2(x_2, y_2) is given by the distance formula: d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) Step 1: Substitute the coordinates of C_1 and C_2. d = sqrt((2-3)^2 + (-4-(-5))^2) Step 2: Calculate the squared differences. d = sqrt((-1)^2 + (1)^2) d = sqrt(1 + 1) d = sqrt(2) The distance between the centres is sqrt(2) units. 5.3 To show that the circles intersect, we need to compare the distance between their centres (d) with the sum and difference of their radii (r_1 and r_2). From 5.1, r_1 = 7. From 5.2, r_2 = sqrt(25) = 5. From 5.2, d = sqrt(2) ≈ 1.414. Step 1: Calculate the sum of the radii. r_1 + r_2 = 7 + 5 = 12 Step 2: Calculate the absolute difference of the radii. |r_1 - r_2| = |7 - 5| = 2 Step 3: Compare d with r_1 + r_2 and |r_1 - r_2|. For intersection, the condition is |r_1 - r_2| < d < r_1 + r_2. 2 < sqrt(2) < 12 This statement is false because 2 > sqrt(2). Therefore, the circles do not intersect. There might be an error in the question's premise or my calculation. Let me recheck. Recheck 5.1: x^2 - 6x + 9 + y^2 + 10y + 25 = 15 + 9 + 25 = 49. Centre (3,-5), radius r_1=7. Correct. Recheck 5.2: Centre (2,-4), radius r_2=5. Distance d = sqrt((2-3)^2 + (-4-(-5))^2) = sqrt((-1)^2 + (1)^2) = sqrt(1+1) = sqrt(2). Correct. The condition for intersection is that the distance between centers is less than or equal to the sum of radii and greater than or equal to the absolute difference of radii. |r_1 - r_2| d r_1 + r_2. |7-5| sqrt(2) 7+5 2 sqrt(2) 12 This is still false because 2 sqrt(2). The circles do not intersect. They are separate. Correction: The question asks to show that they intersect. This implies they should intersect. Let me re-evaluate the condition for intersection. If d < |r_1 - r_2|, one circle is entirely inside the other. If d = |r_1 - r_2|, they touch internally. If |r_1 - r_2| < d < r_1 + r_2, they intersect at two points. If d = r_1 + r_2, they touch externally. If d > r_1 + r_2, they are separate. In our case, d = sqrt(2) ≈ 1.414. |r_1 - r_2| = 2. r_1 + r_2 = 12. Since d < |r_1 - r_2| (i.e., 1.414 < 2), this means one circle is entirely inside the other. Specifically, the smaller circle (radius 5) is inside the larger circle (radius 7). They do not intersect in the sense of crossing each other. They do not even touch. Given the instruction "show that the circles intersect", there might be a typo in the question or the provided equations. However, based on the given equations, the circles do not intersect. One circle is contained within the other. Let's assume the question meant to ask if one circle is contained within the other, or if they touch. Since d = sqrt(2) and r_1 = 7, r_2 = 5. The distance between centers is sqrt(2). The radius of the smaller circle is 5. The distance from C_1 to the edge of C_2 in the direction of C_1 is d - r_2 = sqrt(2) - 5, which is negative. This means C_1 is inside C_2. The distance from C_2 to the edge of C_1 in the direction of C_2 is d - r_1 = sqrt(2) - 7, which is negative. This means C_2 is inside C_1. Let's check which circle is inside which. C_1 = (3, -5), r_1 = 7. C_2 = (2, -4), r_2 = 5. Distance d = sqrt(2). Since d + r_2 = sqrt(2) + 5 ≈ 6.414, and r_1 = 7. Since d + r_2 < r_1 (6.414 < 7), circle 2 is entirely inside circle 1. They do not intersect. I must state the conclusion based on the calculations. Step 1: Radii are r_1 = 7 and r_2 = 5. Step 2: Distance between centres is d = sqrt(2). Step 3: Compare d with r_1 - r_2 and r_1 + r_2. r_1 - r_2 = 7 - 5 = 2 r_1 + r_2 = 7 + 5 = 12 Since d = sqrt(2) ≈ 1.414, we have d < r_1 - r_2 (1.414 < 2). This means that the smaller circle (Circle 2) is entirely contained within the larger circle (Circle 1), and they do not intersect. Therefore, the statement that the circles intersect is incorrect based on the given equations. 5.4 Show that the two circles intersect along the line y = x + 5. Even though the circles do not intersect based on 5.3, I will proceed with finding the equation of the common chord as if they did, to address the question. The common chord equation is found by subtracting the general forms of the two circle equations. Step 1: Write the general form of Circle 1. C_1: x^2 + y^2 - 6x + 10y - 15 = 0 Step 2: Write the general form of Circle 2. The equation is (x-2)^2 + (y+4)^2 = 25. Expand this: x^2 - 4x + 4 + y^2 + 8y + 16 = 25 x^2 + y^2 - 4x + 8y + 20 = 25 C_2: x^2 + y^2 - 4x + 8

Determine the centre and radius of the circle x2 + y2 - 6x + 10y - 15 = 0 and the distance between this circle and another circle (x-2)2 + (y+4)2 = 25.

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Determine the centre and radius of the circle x2 + y2 - 6x + 10y - 15 = 0 and the distance between this circle and another circle (x-2)2 + (y+4)2 = 25.

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