Find the equation of the circle that passes through the point P(7,5) and those whose center is the point C(4,1) than sketch its graph
Here's a graph of the circle: ttps://www.desmos.com/calculator/jjdnddjczx
To find the equation of a circle given the center and a point on the circle, you can use the general equation of a circle. The general equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle, and r is the radius.
In this case, the center of the circle is C(4, 1), and the given point on the circle is P(7, 5). Let's substitute these values into the equation.
First, we need to find the distance between the center C(4, 1) and the point P(7, 5) to determine the radius. We can use the distance formula:
Distance between two points = √[(x2 - x1)^2 + (y2 - y1)^2]
Let's calculate it:
d = √[(7 - 4)^2 + (5 - 1)^2]
= √[3^2 + 4^2]
= √[9 + 16]
= √25
= 5
Now that we know the radius, which is 5, we can substitute the center and the radius into the general equation of the circle:
(x - 4)^2 + (y - 1)^2 = 5^2
(x - 4)^2 + (y - 1)^2 = 25
So, the equation of the circle is (x - 4)^2 + (y - 1)^2 = 25.
To sketch the graph of the circle, plot the center C(4, 1) on a coordinate plane and mark the radius (5 units) in all directions. Connect the points on the circumference of the circle to form a closed curve.
centre is (4,1), so start with
(x-4)^2 + (y-1)^2 = r^2
but (7,5) lies on it, so
(7-4)^2 + (5-1)^2 = r^2
r^2 = 25
(x-4)^2 + (y-1)^2 = 25