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 of your answer is not an interger express in radical form
c. write an equation for the circle
The center is the midpoint of PQ: (-3,4)
The radius is half the length of PQ: 1/2 √(14^2+4^2)
Now, knowing the center and the radius, you know that the equation is
(x+3)^2 + (y-4)^2 = r^2
i got (x+3)^2 + (y-2)^2=65
Please help
To find the center of the circle, we can use the midpoint formula. The formula for finding the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
a. For our given points P(-10, 2) and Q(4, 6), we can find the midpoint as follows:
Midpoint = ((-10 + 4)/2, (2 + 6)/2)
= (-6/2, 8/2)
= (-3, 4)
Therefore, the center of the circle is (-3, 4).
b. To find the radius of the circle, we can use the distance formula. The formula for finding the distance between two points (x1, y1) and (x2, y2) is:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the points P(-10, 2) and Q(4, 6), we can find the distance as follows:
Distance = √((4 - (-10))^2 + (6 - 2)^2)
= √((4 + 10)^2 + (6 - 2)^2)
= √(14^2 + 4^2)
= √(196 + 16)
= √212
Therefore, the radius of the circle is √212 (expressed in radical form).
c. The equation for a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Using the center (-3, 4) and the radius √212, the equation for the circle becomes:
(x + 3)^2 + (y - 4)^2 = (√212)^2
(x + 3)^2 + (y - 4)^2 = 212
So, the equation for the circle is (x + 3)^2 + (y - 4)^2 = 212.
To find the center of the circle, you can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points, (x₁, y₁) and (x₂, y₂) is given by:
Midpoint = ((x₁ + x₂)/2 , (y₁ + y₂)/2)
a. Using the midpoint formula, let's find the center of the circle:
Coordinates of P: x₁ = -10, y₁ = 2
Coordinates of Q: x₂ = 4, y₂ = 6
Center of the circle = ((-10 + 4)/2 , (2 + 6)/2)
= (-6/2 , 8/2)
= (-3 , 4)
Therefore, the center of the circle is (-3, 4).
b. To find the radius of the circle, we can use the distance formula. The distance formula states that the distance between two points, (x₁, y₁) and (x₂, y₂) is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Using the distance formula, let's find the distance between the center of the circle and one of the endpoints, P or Q:
Coordinates of P: x₁ = -10, y₁ = 2
Coordinates of the center: x₂ = -3, y₂ = 4
Distance = √((-3 - (-10))² + (4 - 2)²)
= √((7)² + (2)²)
= √(49 + 4)
= √53
Therefore, the radius of the circle is √53 (expressed in radical form).
c. To write the equation of the circle, we can use the formula:
Standard Equation of a Circle: (x - h)² + (y - k)² = r²
Where (h, k) represents the center of the circle, and r represents the radius.
From part a, we found that the center of the circle is (-3, 4), and from part b, we found that the radius is √53 (expressed in radical form).
Substituting these values into the equation, we get:
(x - (-3))² + (y - 4)² = (√53)²
(x + 3)² + (y - 4)² = 53
Therefore, the equation of the circle is (x + 3)² + (y - 4)² = 53.