A diameter of a circle has endpoints P(-10, -2) and Q(4, 6).
a. Find the center of the circle.
b. Find the radius. If your answer is not an integer, express it in radical form.
c. Write an equation for the circle.
Can someone explain how to do this?
the center of the circle is the midpoint of the diameter: (-3,2)
The length of the diameter is √(14^2+8^2) = √260 = 2√65
so, the radius is √65
That means the equation is
(x+3)^2 + (y-2)^2 = 65
Of course! Here's a step-by-step explanation to help you solve this problem:
a. To find the center of the circle, you can use the midpoint formula. The midpoint formula states that the x-coordinate of the midpoint is the average of the x-coordinates of the endpoints, and the y-coordinate of the midpoint is the average of the y-coordinates. So, calculate the coordinates of the midpoint using the formula:
Midpoint (x, y) = ((x₁ + x₂)/2, (y₁ + y₂)/2)
In this case, the coordinates of the endpoints are P(-10, -2) and Q(4, 6). Plugging in these values into the formula, we get:
Center (x, y) = ((-10 + 4)/2, (-2 + 6)/2)
= (-6/2, 4/2)
= (-3, 2)
Therefore, the center of the circle is C(-3, 2).
b. To find the radius, you can use the distance formula. The distance formula states that the distance between two points (x₁, y₁) and (x₂, y₂) is given by:
Distance = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
In this case, you need to find the distance between the endpoints P(-10, -2) and Q(4, 6). Plugging in these values into the formula, we get:
Distance = sqrt((4 - (-10))² + (6 - (-2))²)
= sqrt((14)² + (8)²)
= sqrt(196 + 64)
= sqrt(260)
Therefore, the radius of the circle is sqrt(260).
c. To write an equation for the circle, you can use the standard form of a circle equation:
(x - h)² + (y - k)² = r²
where (h, k) is the center of the circle and r is the radius. In this case, the center is C(-3, 2) and the radius is sqrt(260). Plugging in these values into the equation, we get:
(x - (-3))² + (y - 2)² = (sqrt(260))²
(x + 3)² + (y - 2)² = 260
This is the equation of the circle.
Certainly! To find the center of the circle, we can use the midpoint formula.
a. The midpoint formula states that the midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the coordinates ( (x₁+x₂)/2, (y₁+y₂)/2 ).
In this case, we have the endpoints P(-10, -2) and Q(4, 6). Using the midpoint formula, we can find the center of the circle:
Center = ( (-10 + 4)/2, (-2 + 6)/2 )
= ( -6/2, 4/2 )
= ( -3, 2 )
Therefore, the center of the circle is (-3, 2).
b. To find the radius, we need to find the distance between the center and one of the endpoints. We can use the distance formula for this.
The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by
Distance = sqrt( (x₂-x₁)² + (y₂-y₁)² )
Using the distance formula, we can find the radius of the circle by finding the distance between the center (-3, 2) and either point P(-10, -2) or Q(4, 6). Let's use point P:
Radius = sqrt( (4 - (-10))² + (6 - (-2))² )
= sqrt( (4 + 10)² + (6 + 2)² )
= sqrt( (14)² + (8)² )
= sqrt( 196 + 64 )
= sqrt( 260 )
Therefore, the radius of the circle is sqrt(260), which is not an integer.
c. Finally, to write an equation for the circle, we use the standard form of the equation for a circle:
(x - h)² + (y - k)² = r²
where (h, k) represents the center of the circle and r represents the radius. By substituting the values we found earlier, the equation for the circle becomes:
(x - (-3))² + (y - 2)² = (sqrt(260))²
Simplifying, we get:
(x + 3)² + (y - 2)² = 260
So, the equation for the circle is (x + 3)² + (y - 2)² = 260.