A circle passes throughthe origin and the point (5/2,1/2) and has 2y-3x=0 as a diameter. Find its equation.
(x-a)^2 + (y-b)^2 = r^2
(5/2-a)^2 + (1/2-b)^2 = r^2
25/4-5a+a^2+1/4-b+b^2 = r^2
but
a^2+b^2 = r^2 for (0,0)
so
26/4 -5a - b = 0
5 a + b = 26/4
center lies on 2y=3x or through(a,b)
2 b = 3 a
so
26/4 = 5 a + (3/2) a
26 = 20 a + 6 a = 26 a
a = 1
b = 3/2
take it from there
To find the equation of the circle, we need to determine the coordinates of its center and the radius.
Given that the diameter of the circle is given by the equation 2y - 3x = 0, we can rewrite it in the form of the equation for a line: y = (3/2)x.
Since a diameter of a circle always passes through the center, we can find the center of the circle by finding the point where the line intersects the perpendicular bisector of the line segment joining the origin and the point (5/2, 1/2).
The midpoint of the line segment joining the origin (0, 0) and the point (5/2, 1/2) can be found by taking the average of their x-coordinates and y-coordinates respectively.
Midpoint coordinates:
x-coordinate = (0 + 5/2) / 2 = 5/4
y-coordinate = (0 + 1/2) / 2 = 1/4
So, the midpoint is M(5/4, 1/4).
To find the slope of the line 2y - 3x = 0, we can rewrite it in slope-intercept form: y = (3/2)x.
The slope of this line is 3/2, and the slope of the perpendicular line is the negative reciprocal, which is -2/3.
Using the point-slope form of a line, we can find the equation of the perpendicular bisector:
(y - 1/4) = (-2/3)(x - 5/4)
Simplifying the equation:
3y - 3/4 = -2x + 5/2
3y = -2x + 5/2 + 3/4
3y = -2x + 13/4
y = (-2/3)x + 13/12
Now, let's find the intersection point of the two lines, (3/2)x and (-2/3)x + 13/12. Setting them equal to each other:
(3/2)x = (-2/3)x + 13/12
(9/6)x = (-4/6)x + 13/12
(13/6)x = 13/12
x = 13/12 * (6/13)
x = 1/2
Substituting this value back into the equation of the line, we can find the y-coordinate:
y = (-2/3)(1/2) + 13/12
y = -1/3 + 13/12
y = -4/12 + 13/12
y = 9/12
y = 3/4
Thus, the center of the circle is C(1/2, 3/4).
To find the radius, we can use the distance formula between the center of the circle and any point on its circumference. Let's use the point (5/2, 1/2) on the circle:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((5/2 - 1/2)^2 + (1/2 - 3/4)^2)
d = sqrt((4/2)^2 + (-1/4)^2)
d = sqrt(4 + 1/16)
d = sqrt(64/16 + 1/16)
d = sqrt(65/16)
d = sqrt(65)/4
So, the radius of the circle is sqrt(65)/4.
Finally, we can write the equation of the circle using the center and radius:
(x - 1/2)^2 + (y - 3/4)^2 = (sqrt(65)/4)^2
Simplifying the equation:
(x - 1/2)^2 + (y - 3/4)^2 = 65/16
Therefore, the equation of the circle is (x - 1/2)^2 + (y - 3/4)^2 = 65/16.
To find the equation of the circle, we need to find its center and radius.
Step 1: Find the center of the circle.
The center of a circle is the midpoint of its diameter. Since the diameter of the circle is given by the equation 2y - 3x = 0, we need to find the midpoint of this line.
Let's write the equation in the form y = mx + b:
2y - 3x = 0
2y = 3x
y = (3/2)x
Now we can find the midpoint by taking the average of the x-coordinates and the average of the y-coordinates of the two points on the line.
Let's call the coordinates of the center of the circle (h, k).
Average of x-coordinates: (0 + 5/2) / 2 = 5/4
Average of y-coordinates: (0 + 1/2) / 2 = 1/4
So the center of the circle is (h, k) = (5/4, 1/4).
Step 2: Find the radius of the circle.
The radius of a circle is the distance from its center to any point on the circle. We can use the distance formula to find the distance between the center of the circle and the point (5/2, 1/2).
The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates, we have:
d = sqrt((5/2 - 5/4)^2 + (1/2 - 1/4)^2)
= sqrt((5/4)^2 + (1/4)^2)
= sqrt(25/16 + 1/16)
= sqrt(26/16)
= sqrt(13/8)
= sqrt(13)/sqrt(8)
Step 3: Write the equation of the circle.
The general equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Substituting the values we have:
(x - 5/4)^2 + (y - 1/4)^2 = (sqrt(13)/sqrt(8))^2
(x - 5/4)^2 + (y - 1/4)^2 = 13/8
Therefore, the equation of the circle is (x - 5/4)^2 + (y - 1/4)^2 = 13/8.
At any point (x, 3/2 x) on the line, the distance from the origin is
d^2 = x^2 + 9/4 x^2 = 13/4 x^2
If the circle's center is at (h,3h/2), then, we have
(x-h)^2 + (y-(3h/2))^2 = 13/4 h^2
(5/2-h)^2 + (1/2-(3h/2))^2 = 13/4 h^2
h=1
So, the circle is
(x-1)^2 + (y-3/2)^2 = 13/4
You can see from the plots below that the circle meets the requirements