find points on the curve x^2+xy+y^2=7

(a) where the tangen is parallel to the x-axis and

(b) wehre the tangent is parallel to athe y-axis.

(in this case, dy/dx is not defined, but dx/dy is. what value does dx/dy have at these point?)

(a) Points where the tangent is parallel to the x-axis can be found by setting dy/dx = 0 in the equation x^2 + xy + y^2 = 7. Let's calculate dy/dx first:

Differentiating x^2 + xy + y^2 = 7 implicitly with respect to x, we get:
2x + (x)(dy/dx) + y + 2y(dy/dx) = 0

Rearranging the terms, we have:
(2x + y) + (x + 2y)(dy/dx) = 0

Now, setting dy/dx = 0 yields:
2x + y = 0

Substituting for y, we have:
2x + (-2x) = 0
0 = 0

From this equation, we can see that for any value of x, y = -2x. So, the points on the curve where the tangent is parallel to the x-axis are given by (x, -2x).

(b) Points where the tangent is parallel to the y-axis can be found by setting dx/dy = 0 in the equation x^2 + xy + y^2 = 7. Let's calculate dx/dy:

Differentiating x^2 + xy + y^2 = 7 implicitly with respect to y, we get:
2x(dy/dy) + x + 2y(dy/dy) = 0

Simplifying the equation, we have:
(x + 2y)(dy/dy) = -x

Setting dx/dy = 0, we have -x = 0. This equation implies x must equal 0.

Now, substituting x = 0 back into the original equation, we get:
0 + 0 + y^2 = 7
y^2 = 7

Therefore, there are no points on the curve where the tangent is parallel to the y-axis.

As for the value of dx/dy at these points, since in case (b) dy/dx is not defined, dx/dy is inconsequential because dy/dy = 1. Hence, dx/dy is not applicable in this scenario.

To find points on the curve where the tangent is parallel to the x-axis, we need to find the values of x and y that satisfy the equation and simultaneously make the derivative dy/dx equal to 0.

Step 1: Differentiate the equation with respect to x,
d/dx (x^2 + xy + y^2) = d/dx(7)

Step 2: Using the chain rule and product rule, we get,
2x + x(dy/dx) + y + 2y(dy/dx) = 0

Step 3: Since we want the tangent to be parallel to the x-axis, dy/dx should be equal to 0. Thus, the equation becomes,
2x + xy + y = 0

Step 4: Now we have a system of two equations: x^2 + xy + y^2 = 7 and 2x + xy + y = 0.

To find points on the curve where the tangent is parallel to the y-axis, we need to find the values of x and y that satisfy the equation and simultaneously make the derivative dx/dy equal to 0.

Step 1: Differentiate the equation with respect to y,
d/dy (x^2 + xy + y^2) = d/dy(7)

Step 2: Using the chain rule and product rule, we get,
2y + x + x(dy/dx) + 2y(dy/dx) = 0

Step 3: The derivative dx/dy is defined in this case, so we can calculate it. Rearranging the equation to solve for dx/dy, we have,
dx/dy = -(x + 2y)/(2x + y)

Step 4: Since we want the tangent to be parallel to the y-axis, dx/dy should be equal to 0. Thus, the equation becomes,
-(x + 2y)/(2x + y) = 0

Now, solving the system of equations obtained in steps 3 and 4 will give us the values of x and y at the points where the tangents are parallel to the x-axis and y-axis, respectively.

To find points on the curve where the tangent is parallel to the x-axis, we need to determine where the derivative of y with respect to x is equal to zero. Let's go through the steps:

Step 1: Find the derivative of the given equation with respect to x.
Differentiating both sides of the equation x^2 + xy + y^2 = 7 with respect to x, we get:
2x + x(dy/dx) + y + 2y(dy/dx) = 0

Step 2: Collect terms involving dy/dx:
(x + 2y)(dy/dx) = -2x - y

Step 3: Solve for dy/dx:
Dividing both sides by (x + 2y), we obtain:
dy/dx = (-2x - y) / (x + 2y)

To find points on the curve where the tangent is parallel to the y-axis, we need to determine where the derivative of x with respect to y is equal to zero. However, given that dy/dx is not defined, we need to find dx/dy instead. Let's proceed with the steps:

Step 1: Find the derivative of the given equation with respect to y.
Differentiating both sides of the equation x^2 + xy + y^2 = 7 with respect to y, we get:
x(dx/dy) + y + x(dy/dy) + 2y(dy/dy) = 0

Step 2: Collect terms involving dx/dy:
x(dx/dy) + x(dy/dy) + 2y(dy/dy) = -y

Step 3: Solve for dx/dy:
Dividing both sides by x + 2y, we obtain:
dx/dy = -y / (x + y)

To find the value of dx/dy at these points, substitute the x and y values of the respective points into the derived equation for dx/dy.