SOLUTION: A circle has equation x^2+y^2-2x+4y-15=0 Find the values for which the line mx+2y-7=0 is a tangent to this circle

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Question 966011: A circle has equation x^2+y^2-2x+4y-15=0
Find the values for which the line mx+2y-7=0 is a tangent to this circle
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
A circle has equation x^2+y^2-2x+4y-15=0
Find the values for which the line mx+2y-7=0 is a tangent to this circle
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x^2+y^2-2x+4y-15=0
(x-1)^2 + (y+2)^2 = 20 is the circle.
mx+2y-7=0
y = mx/2 + 3.5
A point on the tangent lines (2 of them) is A(0,3.5)
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The center of the circle is C(1,-2)
The distance AC =
The tangent point is P
CP = r = sqrt(20)
ACP is a right triangle

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Draw a circle centered at A with r^2 = 11.25
--> x^2 + (y-3.5)^2 = 11.25
The 2 points of intersection of the 2 circles are the tangent points.
x^2 + (y-3.5)^2 = 11.25
x^2 + y^2 - 7y + 12.25 = 11.25
x^2 + y^2 - 7y = -1
x^2 + y^2 - 2x + 4y = 15
------------------------------------------ Subtract
2x - 11y = -16 (the eqn of the line thru the 2 tangent points)
Find the intersection of the line and either circle.
2x - 11y = -16
y = (2x + 16)/11
(x-1)^2 + (y+2)^2 = 20
Sub for y
(x-1)^2 + ((2x+38)/11)^2 = 20
121x^2 - 242x + 121 + 4x^2 + 152x + 1444 = 2420
125x^2 - 90x - 855 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 125x%5E2%2B-90x%2B-855+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-90%29%5E2-4%2A125%2A-855=435600.

Discriminant d=435600 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--90%2B-sqrt%28+435600+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-90%29%2Bsqrt%28+435600+%29%29%2F2%5C125+=+3
x%5B2%5D+=+%28-%28-90%29-sqrt%28+435600+%29%29%2F2%5C125+=+-2.28

Quadratic expression 125x%5E2%2B-90x%2B-855 can be factored:
125x%5E2%2B-90x%2B-855+=+%28x-3%29%2A%28x--2.28%29
Again, the answer is: 3, -2.28. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+125%2Ax%5E2%2B-90%2Ax%2B-855+%29

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x = -2.28 --> y = 1.04 --> (-2.28,1.04)
x = 3 --> y = 2 --> (3,2)
Can you do the rest?