Answer: the minimum distance from P to the circle is approximately 2.97, and the maximum distance is approximately 3.89.
Step-by-step explanation:
To find the minimum and maximum distances from the point P(-3, 9) to the circle defined by (x-3)^2 + (y-1)^2 = 25, we can use the fact that these distances are given by the perpendiculars from the point P to the line passing through the center of the circle.
The center of the circle is (3,1), so we can find the equation of the line passing through P and the center of the circle as follows:
The slope of the line passing through P and the center of the circle is (1-9)/(3-(-3)) = -8/6 = -4/3.
Using the point-slope form of a line, the equation of the line passing through P and the center of the circle is y - 9 = (-4/3)(x + 3).
Now we can find the points where this line intersects the circle. Substituting y = (-4/3)(x+3) + 9 into the equation of the circle, we get:
(x-3)^2 + ((-4/3)(x+3) + 8)^2 = 25
Expanding and simplifying this equation gives a quadratic equation in x:
25x^2 + 96x + 80 = 0
Solving this quadratic equation using the quadratic formula, we get:
x = (-96 ± sqrt(96^2 - 42580)) / (2*25)
x = (-96 ± 56) / 50
x = -2.04 or x = -1.52
Substituting these values of x into y = (-4/3)(x+3) + 9 gives the corresponding values of y:
When x = -2.04, y = 6.24
When x = -1.52, y = 7.27
So the two points of intersection are approximately (-2.04, 6.24) and (-1.52, 7.27).
Finally, we can find the distances from P to each of these points using the distance formula:
The distance from P to (-2.04, 6.24) is sqrt[(-3 - (-2.04))^2 + (9 - 6.24)^2] ≈ 3.89.
The distance from P to (-1.52, 7.27) is sqrt[(-3 - (-1.52))^2 + (9 - 7.27)^2] ≈ 2.97.
Therefore, the minimum distance from P to the circle is approximately 2.97, and the maximum distance is approximately 3.89.