Answer:
The equation in y = mx+b form is y = (4/3)x - 20/3
The equation in Ax+By = C form is 4x - 3y = 20
There are other ways to express the second equation.
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Explanation:
First we must complete the square for x and y.
This will get the equation into (x-h)^2+(y-k)^2 = r^2 form so we can determine the center.
x^2+y^2 - 8x-14y + 40 = 0
(x^2-8x) + (y^2-14y) + 40 = 0
(x^2-8x+16 - 16) + (y^2-14y+49-49) + 40 = 0
(x^2-8x+16) + (y^2-14y+49) + (-16-49+40) = 0
(x-4)^2 + (y-7)^2 = 25
(x-4)^2 + (y-7)^2 = 5^2
This circle is centered at (h,k) = (4,7) and it has radius r = 5.
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Next, we need to find the slope of line AB such that
A = (4,7) = center of the circle
B = (8,4) = point on the circle, which is the point of tangency
m = slope
m = (y2-y1)/(x2-x1)
m = (4-7)/(8-4)
m = -3/4
The slope of line AB is -3/4. Any perpendicular slope will have us do two things:
- flip the fraction
- flip the sign
that would get us from -3/4 to +4/3 or just 4/3.
The perpendicular slope is 4/3.
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So we must find the equation of the line with slope m = 4/3 and it goes through (x,y) = (8,4)
That means,
y = mx+b
4 = (4/3)*8 + b
4 = 32/3 + b
4 - 32/3 = b
12/3 - 32/3 = b
(12-32)/3 = b
-20/3 = b
b = -20/3
Therefore, the equation of the tangent line is y = (4/3)x - 20/3
Optionally, we can get this equation into standard form like so
y = (4/3)x - 20/3
3y = 4x - 20 ... multiply everything by 3
3y-4x = -20
-4x+3y = -20
4x - 3y = 20
The diagram is shown below.
