Find the equation of the circle which is concentric with the circle x^2+y^2+3y-8y+16=0 and tangent to the line 4x+3y-12=0. Show your solution.
find the distance from the center of the circle to the line. Use that for the radius of the circle.
x^2+y^2+3x-8y+16=0 is the circle
(x + 3/2)^2 + (y-4)^2 = 9/4
So, the center of the circle is at (-3/2,4)
The distance to the line 4x+3y-12=0 is thus
|4(-3/2) + 3(4) - 12|/√(4^2+3^2) = 6/5
So, the desired circle is
(x + 3/2)^2 + (y-4)^2 = 36/25
To find the equation of the circle that is concentric with the given circle and tangent to the line, we will follow these steps:
1. Rewrite the equation of the given circle in standard form.
2. Find the center and radius of the given circle.
3. Determine the distance between the center of the given circle and the line.
4. Calculate the radius of the new circle.
5. Write the equation of the new circle using the center and radius.
Let's begin by solving each step:
1. Rewrite the equation of the given circle in standard form:
x^2 + y^2 + 3y - 8y + 16 = 0
Combine like terms:
x^2 - 5y + 16 = 0
2. Find the center and radius of the given circle:
Rewrite the equation in the general form of a circle: (x - h)^2 + (y - k)^2 = r^2
x^2 - 5y + 16 = 0
Complete the square for y:
x^2 - 5y + 25/4 - 25/4 + 16 = 0
(x^2 + (25/4)) - 5(y - (5/2)) = -16 + 25/4
(x - 0)^2 + [y - (5/2)]^2 = 9/4
Comparing this with the standard form, we find that the center is at (0, 5/2) and the radius is 3/2.
3. Determine the distance between the center of the given circle and the line:
The distance between a point (x1, y1) and a line Ax + By + C = 0 is given by the formula:
d = |Ax1 + By1 + C| / sqrt(A^2 + B^2)
Plugging the values of the point (0, 5/2) and the line 4x + 3y - 12 = 0 into the formula:
d = |4(0) + 3(5/2) - 12| / sqrt(4^2 + 3^2)
d = |0 + 15/2 - 12| / sqrt(16 + 9)
d = |15/2 - 24/2| / sqrt(25)
d = |-9/2| / 5
d = 9/10
4. Calculate the radius of the new circle:
The radius of the new circle will be the distance between the center of the given circle and the line, minus the radius of the given circle:
r_new = (3/2) - (9/10)
r_new = (15/10) - (9/10)
r_new = 6/10
r_new = 3/5
5. Write the equation of the new circle using the center and radius:
The equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values, the equation of the new circle is:
(x - 0)^2 + (y - (5/2))^2 = (3/5)^2
Simplifying, the equation becomes:
x^2 + (y - 5/2)^2 = 9/25
Thus, the equation of the circle that is concentric with the given circle and tangent to the line is x^2 + (y - 5/2)^2 = 9/25.