A circle passes through origin and the point (5/2, 1/2) and has 2y_3x=0 as a diameter. Find its equation.
The center lies on the perpendicular bisector of the chord joining (0,0) and (5/2,1/2). That chord has
slope = 1/5
midpoint = (5/4,1/4)
So, the bisector is the line
y - 1/4 = -5(x - 5/4)
The center of the circle is at the intersection of the bisector and the line 2y=3x.
So, the center is at (1,3/2)
Since it passes through (0,0), r^2 = 13
That means the circle is
(x-1)^2 + (y-3/2)^2 = 13/4
see the graph at
http://www.wolframalpha.com/input/?i=plot+(x-1)%5E2+%2B+(y-3%2F2)%5E2+%3D+13%2F4,+2y%3D3x,x%3D5%2F2,y%3D1%2F2
or, you know that
(0-h)^2 + (0-k)^2 = r^2
(5/2-h)^2 + (1/2-k)^2 = r^2
2k=3h
solve those and you get the three values.
To find the equation of the circle, we need to find its center and radius.
First, we need to find the center of the circle. Since the circle passes through the origin, the center of the circle will be at the midpoint between the origin and the given point (5/2, 1/2).
Using the midpoint formula, we can find the center as follows:
x-coordinate of the center = (0 + 5/2) / 2 = 5/4
y-coordinate of the center = (0 + 1/2) / 2 = 1/4
So, the center of the circle is (5/4, 1/4).
Next, we need to find the radius of the circle. We are given that the diameter of the circle is given by the equation 2y - 3x = 0.
The radius of the circle is half the length of the diameter. To find the radius, we need to calculate the distance between the center of the circle and any point on the diameter.
Let's consider the point (0, 0) on the diameter. Using the distance formula, we can find the radius as follows:
radius = sqrt((5/4 - 0)^2 + (1/4 - 0)^2)
= sqrt((5/4)^2 + (1/4)^2)
= sqrt(25/16 + 1/16)
= sqrt(26/16)
= sqrt(13/8)
= sqrt(13)/sqrt(8)
= sqrt(13)/(2sqrt(2))
So, the radius of the circle is sqrt(13)/(2sqrt(2)).
Now, we can write the equation of the circle in the standard form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
Plugging in the values we found, the equation of the circle is:
(x - 5/4)^2 + (y - 1/4)^2 = (sqrt(13)/(2sqrt(2)))^2
Simplifying further,
(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), with 2y - 3x = 0 as a diameter, is (x - 5/4)^2 + (y - 1/4)^2 = 13/8.