A circle passes through the origin and the point (5/2,1/2) and has 2y-3x=0 as a diameter find its equation
To find the equation of a circle, we need to determine its center and radius. Given that the circle passes through the origin and the point (5/2, 1/2), we can find the center of the circle by finding the midpoint of the segment connecting these two points.
The midpoint formula is:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Using the given points, we can calculate the coordinates of the midpoint as follows:
Midpoint = ((0 + 5/2)/2, (0 + 1/2)/2)
Midpoint = ((5/4), (1/4))
So, the center of the circle is (5/4, 1/4).
Next, we need to find the radius of the circle. Since the line 2y - 3x = 0 is a diameter of the circle, we can find the distance between the center of the circle and any point on the line to determine the radius.
We can choose either of the given points, say (5/2, 1/2), to calculate the radius using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the center (5/4, 1/4) and the point (5/2, 1/2), we get:
Distance = √(((5/2) - (5/4))^2 + ((1/2) - (1/4))^2)
Distance = √((5/4)^2 + (1/4)^2)
Distance = √(25/16 + 1/16)
Distance = √(26/16)
Distance = √(13/8)
Distance = √(13)/√(8)
So, the radius of the circle is √(13)/√(8).
Finally, we can write the equation of the circle using the center and radius:
Equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
Substituting the values of the center (h, k) and radius r, we have:
(x - 5/4)^2 + (y - 1/4)^2 = (√13/√8)^2
Simplifying the equation, we get:
(x - 5/4)^2 + (y - 1/4)^2 = 13/8
Therefore, the equation of the circle passing through the origin and the point (5/2, 1/2), and with 2y - 3x = 0 as a diameter is:
(x - 5/4)^2 + (y - 1/4)^2 = 13/8