Find the equation of the circle tangent to the line 2x - 3y + 2 = 0 at (5,4) and with radius equal to √10
so wrong!
2x - 3y + 2 = 0 is never tangent to the circle
(x - 5)^2 + (y - 4)^2 = 10
The slope of the radius from the center (h,k) of the circle to the point (5,4) is -3/2 So, if (h,k) = (5-2z,4+3z), then we have
√13 z = √10
z = √(10/13)
so the center is at (5-2√(10/13) , 4+3√(10/13))
That makes the equation of the circle
(x - (5-2√(10/13)))^2 + (y - (4+3√(10/13)))^2 = 10
Apologies for the mistake made in the previous response.
The equation of the circle tangent to the line 2x - 3y + 2 = 0 at (5,4) and with a radius equal to √10 is actually:
(x - (5-2√(10/13)))^2 + (y - (4+3√(10/13)))^2 = 10
To find the equation of a circle tangent to a given line at a specific point, we can follow these steps:
Step 1: Determine the distance between the center of the circle and the line.
Step 2: Find the equation of the line perpendicular to the given line passing through the point of tangency.
Step 3: Determine the coordinates of the center of the circle.
Step 4: Use the center and radius to write the equation of the circle.
Let's start with step 1. The distance between a point (x1, y1) and a line Ax + By + C = 0 is given by the formula:
d = |Ax1 + By1 + C| / √(A^2 + B^2)
In this case, the given line is 2x - 3y + 2 = 0, and the point of tangency is (5, 4). So, we have:
d = |2(5) - 3(4) + 2| / √(2^2 + (-3)^2)
= |10 - 12 + 2| / √4 + 9
= |-4| / √13
= 4 / √13
Therefore, the distance between the center of the circle and the line is 4 / √13.
Moving on to step 2, we need to find the equation of the line perpendicular to the given line, passing through the point (5, 4). For two lines to be perpendicular, the product of their slopes should be -1.
The slope of the given line is found by rearranging the equation into slope-intercept form (y = mx + b), where m is the slope:
2x - 3y + 2 = 0
-3y = -2x - 2
y = (2/3)x + 2/3
The slope of the given line is 2/3, so the slope of the line perpendicular to it will be -3/2 (the negative reciprocal).
Using the point-slope form of a line (y - y1 = m(x - x1)), we can write the equation of the line perpendicular to the given line passing through (5, 4):
y - 4 = (-3/2)(x - 5)
y - 4 = (-3/2)x + 15/2
y = (-3/2)x + 23/2
Now, in step 3, we need to determine the coordinates of the center of the circle. Since the center of the circle lies on the perpendicular line, it must satisfy its equation. So, we substitute y = (-3/2)x + 23/2 into the equation of the line:
(-3/2)x + 23/2 = (-3/2)x + 23/2
This means the equation is true for any value of x. Hence, the center of the circle can have any x-coordinate but must have a y-coordinate of (-3/2)x + 23/2.
Lastly, in step 4, we can use the center and the given radius (√10) to write the equation of the circle. The general equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) represents the center of the circle, and r is the radius. In our case, the center is (x, (-3/2)x + 23/2), and the radius is √10. Thus, the equation of the circle is:
(x - x)^2 + (y - (-3/2)x + 23/2)^2 = (√10)^2
(y - (-3/2)x + 23/2)^2 = 10
Simplifying further would require additional information about the specific value of x or the elimination of the variables.
To find the equation of the circle, we first need to find the coordinates of its center.
Since the circle is tangent to the line, the center of the circle must lie on the perpendicular line that passes through the point of tangency (5, 4).
The given line has a slope of 2/3, so the perpendicular line will have a slope equal to the negative reciprocal of 2/3, which is -3/2.
Using the point-slope form of a line, the equation of the perpendicular line passing through (5, 4) is:
y - 4 = -3/2(x - 5)
Simplifying, we get:
y - 4 = -3/2x + 15/2
y = -3/2x + 15/2 + 4
y = -3/2x + 23/2
To find the coordinates of the center of the circle, we solve the system of equations consisting of the given line and the perpendicular line:
2x - 3y + 2 = 0
y = -3/2x + 23/2
Substituting -3/2x + 23/2 for y in the first equation, we get:
2x - 3(-3/2x + 23/2) + 2 = 0
2x + 9/2x - 69/2 + 2 = 0
4x + 9x - 69 + 4 = 0
13x - 65 = 0
13x = 65
x = 5
Substituting x = 5 into y = -3/2x + 23/2, we get:
y = -3/2(5) + 23/2
y = -15/2 + 23/2
y = 8/2
y = 4
Therefore, the center of the circle is (5, 4).
Since the radius of the circle is √10, the equation of the circle is:
(x - 5)^2 + (y - 4)^2 = (√10)^2
(x - 5)^2 + (y - 4)^2 = 10