find the points on the curve [(x^2 )+ (y^2) = 18] at which the tangents are parallel to the line x+y = 3

the line has slope -1

x^2+y^2=18
dy/dx = -x/y

so, we need x = y

x^2+x^2 = 18
x = ±3, so the points are

(-3,-3) and (3,3)

visit wolframalpha.com and enter

plot x^2+y^2=18, x+y = -6, x+y=6, x=3, y=3

To find the points on the curve where the tangents are parallel to the line x + y = 3, we need to find the points where the slopes of the tangent lines are equal to the slope of the line x + y = 3.

Step 1: Rewrite the equation of the line x + y = 3 in slope-intercept form.
Subtract x from both sides: y = -x + 3.

Step 2: Calculate the slope of the line x + y = 3.
The slope-intercept form of a line is y = mx + b, where m is the slope. Comparing with x + y = 3, we have m = -1.

Step 3: Differentiate the equation of the curve with respect to x to find the slope of the tangent line at each point.
Differentiating both sides of the equation (x^2) + (y^2) = 18 with respect to x:
2x + 2y * dy/dx = 0.

Step 4: Solve the equation 2x + 2y * dy/dx = 0 for dy/dx.
Rearranging the equation, we have dy/dx = -2x / (2y).

Step 5: Set the slope of the tangent line equal to the slope of the line x + y = 3 to find the points.
Setting dy/dx = -1, we have -2x / (2y) = -1.

Step 6: Solve the equation -2x / (2y) = -1 for x and y.
Cross-multiplying, we get -2x = -2y. Dividing by -2, we have x = y.

Step 7: Substitute x = y back into the equation of the curve (x^2) + (y^2) = 18 to find the values of x and y.
Substituting x = y into (x^2) + (y^2) = 18, we have y^2 + y^2 = 18, which simplifies to 2y^2 = 18.
Dividing by 2, we get y^2 = 9, and taking the square root gives y = ±3.

Step 8: Substitute the values of y back into the equation x = y to find the corresponding values of x.
If y = 3, then x = 3.
If y = -3, then x = -3.

Therefore, the points on the curve where the tangents are parallel to the line x + y = 3 are (3, 3) and (-3, -3).

To find the points on the curve where the tangents are parallel to the given line, we need to find the derivative of the curve and set it equal to the derivative of the given line. Here are the steps to do that:

1. Start with the equation of the curve: x^2 + y^2 = 18.
2. Differentiate both sides with respect to x to find the derivative of the curve.
d/dx [x^2 + y^2] = d/dx [18]
2x + 2y * dy/dx = 0
3. Solve the above equation for dy/dx (the derivative of y with respect to x):
2y * dy/dx = -2x
dy/dx = -2x / 2y
dy/dx = -x / y
4. Now, we need to find the derivative of the given line x + y = 3.
Differentiate both sides with respect to x:
d/dx [x + y] = d/dx [3]
1 + dy/dx = 0
dy/dx = -1
5. Equate the derivative of the curve (-x / y) to the derivative of the line (-1):
-x / y = -1
x / y = 1
x = y
6. Substitute this value of x in the equation of the curve: x^2 + y^2 = 18.
y^2 + y^2 = 18
2y^2 = 18
y^2 = 9
y = ±3
7. Substitute the values of y back into the equation x = y to find the corresponding x-values:
For y = 3, x = 3.
For y = -3, x = -3.

Therefore, the points on the curve where the tangents are parallel to the line x + y = 3 are (3, 3) and (-3, -3).