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Answer:
y = -4/3x +23/2
Step-by-step explanation:
The tangent of interest is perpendicular to the radius at the given point on the circle. To find the perpendicular, it helps to know the slope of the radius segment. To find that, we need the center of the circle.
The center of the circle can be found by completing the square for each variable.
(x^2 -2x) +(y^2 +4y) = 20
(x^2 -2x +1) +(y^2 +4y +4) = 20 +1 +4
(x -1)^2 +(y +2)^2 = 25
Comparing this equation to the standard form equation for a circle, we can find the center.
(x -h)^2 +(y -k)^2 = r^2 . . . . . . . circle of radius r centered at (h, k)
The center of circle P is (1, -2).
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The slope m' of the line segment between (1, -2) and (5, 1) is given by ...
m' = (y2 -y1)/(x2 -x1)
m' = (1 -(-2))/(5 -1) = 3/4
The slope m of the perpendicular line is the opposite reciprocal of this, so is ...
m = -1/m' = -1/(3/4) = -4/3
The y-intercept of the desired line can be found from ...
b = y -mx . . . . . . . for m=-4/3 and (x, y) = (5, 1)
b = 1 -(-4/3)(5) = 23/3
Then the equation of tangent line AB is ...
y = mx +b
y = -4/3x +23/3
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