A circle C passes through the point (-12,9) and is given by the equation (x-a)^2 + (y-b)^2 = r^2. If the equation of the tangent to the given circle at the point (-4,1) is given by x-y+5=0, find the values of a,b and r.
The answers have to be a=-8, b=5 r=5.657
The center of the circle is at x=a, y = b. It is also on a line that is perpendicular to the tangent line. That line has slope -1 and its equation is
(y-1) = -(x+4)
y = -x -3, so since (a,b) is on that line,
b = -a -3
Another equation that relates a to be is
(-12-a)^2 + (9-b)^2
= (-4-a)^2 + (1-b)^2 = r^2
144 +24a + a^2 +81-18b+b^2 = 16 +8a +a^2 +b^2-2b +1
208 +16a -16 b = 0
a - b +13 = 0
Your two equations for a and b can now be solved to get a = -8; b = 5
Insert those values into the original circle equation using any known point on the circle,to get r
(-4+8)^2 + (1-5)^2 = r^2
r = sqrt (16 + 16) = 5.657
thanks!
I should have written:
"(a,b) is also on a line that is perpendicular to the tangent line and passes through its point of contact with the circle", and
"Another equation that relates a to b is.."
The derivation remains the same
To find the values of a, b, and r, we need to solve the given equations and find the common points of the circle and the tangent line.
Step 1: Equation of the Circle
The equation of the circle is given as (x - a)^2 + (y - b)^2 = r^2. We need to substitute the given point (-12, 9) into this equation to obtain one equation.
(-12 - a)^2 + (9 - b)^2 = r^2
Step 2: Equation of the Tangent Line
The equation of the tangent line is given as x - y + 5 = 0. We need to find the point of tangency by solving this equation.
x - y + 5 = 0
Step 3: Find the point of tangency
To find the point of tangency, we need to solve the equations of the circle and the tangent line simultaneously. We have:
(-12 - a)^2 + (9 - b)^2 = r^2
x - y + 5 = 0
Step 4: Substitute the coordinates of the point of tangency
Substitute the coordinates of the point of tangency (-4, 1) into the equation of the circle and the equation of the tangent line.
(-12 - a)^2 + (9 - b)^2 = r^2
-4 - 1 + 5 = 0
Step 5: Solve the system of equations
Solve the system of equations obtained in Step 4 to find the values of a, b, and r.
From the second equation, we have:
0 = 0
This equation is always true and does not provide any additional information about a, b, and r.
From the first equation, we have:
(-12 - a)^2 + (9 - b)^2 = r^2
By substituting the coordinates (-4, 1) into this equation, we get:
(-12 - a)^2 + (9 - b)^2 = r^2
(-12 + 4 - a)^2 + (9 - 1 - b)^2 = r^2
(-8 - a)^2 + (8 - b)^2 = r^2
64 + 16a + a^2 + 64 + 16b + b^2 = r^2
a^2 + b^2 + 16a + 16b + 128 = r^2
Now, substitute the given values of a, b, and r into this equation:
(-8)^2 + 5^2 + 16(-8) + 16(5) + 128 = r^2
64 + 25 - 128 + 80 + 128 = r^2
169 = r^2
Taking the square root of both sides, we get:
r = ±√169
r = ±13
So, we have two possible values for r: r = 13 and r = -13.
However, since a circle cannot have a negative radius, we take r = 13.
Step 6: Find the values of a and b
Now that we have r = 13, substitute this value back into the equation of the circle to find the values of a and b.
(-12 - a)^2 + (9 - b)^2 = (13)^2
(-12 - a)^2 + (9 - b)^2 = 169
Since the equation of the tangent line is x - y + 5 = 0, we can rewrite it as y = x + 5.
Substituting this into the circle equation, we get:
(-12 - a)^2 + (9 - (a + 5))^2 = 169
(-12 - a)^2 + (9 - a - 5)^2 = 169
(-12 - a)^2 + (4 - a)^2 = 169
144 + 24a + a^2 + 16 - 8a + a^2 = 169
2a^2 + 16a + 160 = 169
2a^2 + 16a - 9 = 0
Solving this quadratic equation, we find:
a = -8 or a = 0.5625
Since we know that a = -8, we can substitute this back into the equation to find b:
(-12 - (-8))^2 + (9 - b)^2 = 169
(-12 + 8)^2 + (9 - b)^2 = 169
16^2 + (9 - b)^2 = 169
256 + 81 - 18b + b^2 = 169
b^2 - 18b + 337 = 0
Solving this quadratic equation, we get:
b = 5 or b = 13
Since we have already determined that the circle passes through the point (-12, 9), we can conclude that the values of a, b, and r are a = -8, b = 5, and r = 13.
Therefore, the values of a, b, and r that satisfy the given conditions are a = -8, b = 5, and r = 13.