find the equation of a circle passing through the points p(0,-3) and q(4,0) and has its centre on the line x+2y=0
The circle's equation is
(x-h)^2 + (y-k)^2 = r^2
You know that
(0-h)^2 + (-3-k)^2 = r^2
(4-h)^2 + (0-k)^2 = r^2
h+2k = 0
Now solve for h,k,r
I'd start with
h^2+(k+3)^2 = (h-4)^2+k^2
h = -2k, so
4k^2 + (k+3)^2 = (-2k-4)^2 + k^2
solve that for k, and then you can get h and r.
Thanks for ur help
Thanks a lot steve .... appreciate that !!
But can i use elimination method to solve for h and k ?
sure - do whatever works for you.
To find the equation of a circle passing through two given points and having its center on a given line, we need to follow these steps:
1. Find the midpoint of the line segment connecting the two given points. This will give us the coordinates of the center of the circle.
2. Determine the radius of the circle by finding the distance between one of the given points and the center.
3. Use the center coordinates and the radius to write the equation of the circle.
Let's go through each step one by one.
Step 1: Find the midpoint of the line segment connecting p(0, -3) and q(4, 0).
The midpoint formula is:
Midpoint = ((x1 + x2) / 2 , (y1 + y2) / 2)
Using this formula, we find the midpoint of p(0, -3) and q(4, 0) as follows:
Midpoint = ((0 + 4) / 2 , (-3 + 0) / 2)
= (2, -3/2)
So, the coordinates of the center of the circle are (2, -3/2).
Step 2: Determine the radius of the circle.
To find the radius, we can use the distance formula between the center (2, -3/2) and either of the given points. Let's use point p(0, -3) to find the distance.
The distance formula is:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we calculate the distance between (2, -3/2) and (0, -3) as follows:
Distance = √((0 - 2)^2 + (-3 - (-3/2))^2)
= √((-2)^2 + (-3 + 3/2)^2)
= √(4 + (-1/2)^2)
= √(4 + 1/4)
= √(16/4 + 1/4)
= √(17/4)
= √17 / 2
So, the radius of the circle is √17 / 2.
Step 3: Write the equation of the circle.
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Using the center coordinates (2, -3/2) and the radius √17 / 2, we can write the equation of the circle passing through p(0, -3) and q(4, 0) as follows:
(x - 2)^2 + (y + 3/2)^2 = (√17 / 2)^2
Simplifying the equation further:
(x - 2)^2 + (y + 3/2)^2 = 17/4
Thus, the equation of the circle passing through p(0, -3) and q(4, 0) and having its center on the line x+2y=0 is:
(x - 2)^2 + (y + 3/2)^2 = 17/4