In the xy plane, a circle has center P(5,10) and intersects the x axis at a single point Q(5,0). What is the area of circle P? Express your answer in terms of pi
The radius of the circle is the distance between the center P(5, 10) and the point Q(5, 0) on the x-axis.
Using the distance formula, the radius r is given by:
r = sqrt((5 - 5)^2 + (0 - 10)^2) = sqrt((0)^2 + (-10)^2) = sqrt(100) = 10.
The area of a circle is given by the formula A = πr^2.
Substituting the value of the radius r = 10 into this formula, we get:
A = π(10)^2 = π(100) = 100π.
Therefore, the area of circle P is 100π.
To find the area of a circle, we need to know the radius. In this case, the center of the circle is P(5,10), and it intersects the x-axis at point Q(5,0).
Since point Q lies on the x-axis, its y-coordinate is 0. Therefore, the distance between points P(5,10) and Q(5,0) is equal to the radius of the circle.
Using the distance formula, we can find the radius (r) as follows:
r = √((x2 - x1)^2 + (y2 - y1)^2)
= √((5 - 5)^2 + (0 - 10)^2)
= √(0 + 100)
= √100
= 10
So, the radius of circle P is 10 units.
The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius.
Substituting the radius we found (10) into the formula:
A = π * (10)^2
= π * (100)
= 100π
Therefore, the area of circle P is 100π square units.