find the equation of the circle whose centre lies on the line 5x-2y=9 and the circle touches both the positive axes
The center has to be on the line
y = x
if the circle is tangent to the x and y
axes.
so center on 3 x = 3 y = 9
or
(3,3)
and the radius is 3
so
(x-3)^2 + (y-3)^2 = 9
To find the equation of the circle, we need to determine its center and radius.
Step 1: Find the center of the circle:
Since the center of the circle lies on the line, let's substitute x = 0 and y = 0 into the equation of the line:
5(0) - 2(0) = 9
0 = 9
Since the equation is not satisfied, the center of the circle does not lie on the origin.
Let's solve the equation of the line for x:
5x - 2y = 9
5x = 2y + 9
x = (2y + 9) / 5
Step 2: Substitute the equation of the line into the equation of the circle to solve for the center and radius.
Using the distance formula, the distance from the center (x, y) to each axis is equal to the radius:
Center to x-axis: y
Center to y-axis: x
Since the circle touches both positive axes, the center's coordinates are (r, r), where r is the radius.
Substitute x = (2y + 9) / 5 into x and y coordinates:
x = (2y + 9) / 5
y = (2y + 9) / 5
Simplify:
5x = 2y + 9
5y = 2y + 9
Now, solve for x and y:
5x - 2y = 9
3y = 9
y = 3
Substituting y = 3 back into the equation 5x - 2y = 9:
5x - 2(3) = 9
5x - 6 = 9
5x = 15
x = 3
Therefore, the center of the circle is (3,3).
Step 3: Calculate the radius of the circle:
The radius of the circle is the distance from the center to either the x or y axis.
Using the distance formula:
Radius = sqrt((x - 0)^2 + (y - 0)^2)
Radius = sqrt((3 - 0)^2 + (3 - 0)^2)
Radius = sqrt(9 + 9)
Radius = sqrt(18)
Radius = 3√2
So, the radius of the circle is 3√2.
Step 4: Write the equation of the circle:
Now we know the center and radius of the circle, the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center and r is the radius.
Substituting the values we found:
(x - 3)^2 + (y - 3)^2 = (3√2)^2
(x - 3)^2 + (y - 3)^2 = 18
Thus, the equation of the circle is (x - 3)^2 + (y - 3)^2 = 18.
To find the equation of a circle, we need to know the coordinates of its center and its radius.
In this case, the center of the circle lies on the line 5x - 2y = 9. Let's find the coordinates of the center.
First, let's rearrange the equation 5x - 2y = 9 to solve for y:
-2y = -5x + 9
y = (5/2)x - (9/2)
Now, we have the equation of a line in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Here, the slope is 5/2, which tells us that for every increase of 2 units in x, y increases by 5 units.
Since the circle touches both the positive x-axis and the positive y-axis, we can see that the center of the circle must lie in the first quadrant, where x and y are both positive.
Now, let's consider the radius of the circle. Since the circle touches both the positive x-axis and the positive y-axis, the radius will be the distance from the center of the circle to either of the axes.
Since the center lies on the line with equation 5x - 2y = 9, let's substitute the coordinates of the center into this equation to find the value of 5x - 2y:
5x - 2y = 9
5(x-coordinate) - 2(y-coordinate) = 9
Substituting these values into the equation will give us the value of 5x - 2y, which represents the distance from a point on the circle to the line 5x - 2y = 9.
Let's assume the coordinates of the center are (h, k). Substituting these values into the equation gives us:
5h - 2k = 9
Now we have two equations:
1. 5h - 2k = 9 (from substituting the coordinates of the center into the equation of the line)
2. h^2 + k^2 = r^2 (the equation of a circle centered at (h, k) with radius r)
We want to find the equation of the circle, so we need to find the values of h, k, and r.
Using the fact that the circle touches both positive x-axis and positive y-axis, we know that the radius is equal to the distance from the center to either of these axes.
From geometry, we know that the distance between a point (x, y) and the x-axis is equal to y, and the distance between a point (x, y) and the y-axis is equal to x.
Therefore, the radius can be represented as:
r = h (distance to the y-axis) = k (distance to the x-axis)
Now, we can solve the system of equations to find the values of h, k, and r.
First, solve equation 1 (5h - 2k = 9) for h:
5h = 2k + 9
h = (2k + 9)/5
Substitute this value of h in equation 2 (h^2 + k^2 = r^2):
((2k + 9)/5)^2 + k^2 = r^2
Simplify this equation to find the equation of the circle.