given the circle having the equation x^2+y^2=9, find the shortest distance from the point (4,5). find the longest distance from the point (4,5).

The points on the circle that are the nearest and farthest from (4,5) will be the two points on the line that connects (4,5) with the center of the circle, located at (0,0). The equation of that line is

y = (5/4) x

Points on both the line and the circle must satisfy the equation
x^2 + (25/16)x^2 = 9
(41/16)x^2 = 9
x = + or - 12/sqrt41 = + or - 1.87409
y = (5/4)x = + or - 2.34261
Farthest point: (-1.87409, -2.3261)
Nearest point: (1.87409, 2.3261)

To find the shortest distance from the point (4,5) to the circle with the equation x^2 + y^2 = 9, we need to find the distance between the point and the circle's circumference.

Step 1: Find the equation of the line passing through the center of the circle and the given point.
The center of the circle is at (0,0) since the equation of the circle is x^2 + y^2 = 9. Therefore, the equation of the line passing through the center (0,0) and the point (4,5) is y = (5/4)x.

Step 2: Substitute the equation of the line into the equation of the circle.
Substitute y = (5/4)x into x^2 + y^2 =9:
x^2 + (5/4)x)^2 = 9
x^2 + 25/16x^2 = 9
(16/16)x^2 + (25/16)x^2 = 9
(41/16)x^2 = 9
x^2 = 144/41

Step 3: Solve for x.
x = ± √(144/41)
x ≈ ± 2.45

Step 4: Find the corresponding y-values.
To find the corresponding y-values, substitute the x-values into the equation of the line:
For x = 2.45, y = (5/4)(2.45) ≈ 3.06
For x = -2.45, y = (5/4)(-2.45) ≈ -3.06

Step 5: Calculate the distance between the point (4,5) and the two points on the circle obtained in Step 4.
Using the distance formula, the distance between (4,5) and (2.45, 3.06) is:
√((2.45 - 4)^2 + (3.06 - 5)^2) ≈ √2.88 ≈ 1.695

The shortest distance between the point (4,5) and the circle is approximately 1.695 units.

To find the longest distance, we need to find the distance between the point (4,5) and the farthest point on the circle from the center. Since the farthest point will be on the circumference, it will be one of the outermost points of the circle.

Step 6: Calculate the distance between the point (4,5) and the farthest point on the circle.
The farthest points on the circle from the center are (3, √6) and (-3, -√6). These points are obtained by substituting the boundary values of x into the equation of the circle.

Using the distance formula, the distance between (4,5) and (3, √6) is:
√((3 - 4)^2 + (√6 - 5)^2) ≈ √10.17 ≈ 3.19

Using the distance formula, the distance between (4,5) and (-3, -√6) is:
√((-3 - 4)^2 + (-√6 - 5)^2) ≈ √32.17 ≈ 5.67

Therefore, the longest distance between the point (4,5) and the circle is approximately 5.67 units.

To find the shortest and longest distance between a point and a circle, we need to calculate the distance between the point and the center of the circle, as well as the radius of the circle.

The given equation of the circle is x^2 + y^2 = 9. We can observe that the center of the circle is at the origin (0,0), and the radius is √9 = 3.

Now, let's start with finding the shortest distance from the point (4,5) to the circle:

1. Calculate the distance between the center of the circle and the given point:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((0 - 4)^2 + (0 - 5)^2)
Distance = √((-4)^2 + (-5)^2)
Distance = √(16 + 25)
Distance = √41

2. Subtract the radius of the circle from the calculated distance to get the shortest distance:
Shortest Distance = Distance - Radius
Shortest Distance = √41 - 3

Therefore, the shortest distance from the point (4,5) to the circle x^2 + y^2 = 9 is √41 - 3.

To find the longest distance, we need to calculate the distance between the center of the circle and the given point:

1. Calculate the distance between the center of the circle and the given point:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((0 - 4)^2 + (0 - 5)^2)
Distance = √((-4)^2 + (-5)^2)
Distance = √(16 + 25)
Distance = √41

2. Add the radius of the circle to the calculated distance to get the longest distance:
Longest Distance = Distance + Radius
Longest Distance = √41 + 3

Therefore, the longest distance from the point (4,5) to the circle x^2 + y^2 = 9 is √41 + 3.