Find the points at which the relation 3x^2-2xy+y^2=24 has a vertical or horizontal tangent line.

Differentiate both side with respect to x, implicitly.

6x -2y -2x dy/dx +2y dy/dx = 0
dy/dx*(x-y) = 3x -2y
The tangent is vertical at x = y and horizontal at x = 2y/3. Solve the original equation for the corresponding x and y values.
Horizontal tangent at:
3x^2+2x+x^2 = 24
x^2 +x -12 = 0
x = -4 or 3

To find the points at which a relation has a vertical or horizontal tangent line, we need to find the points where the derivative of the relation with respect to the variable is either zero (for horizontal tangent line) or undefined (for vertical tangent line).

Let's start by finding the partial derivatives of the given Equation. The equation is: 3x^2 - 2xy + y^2 = 24.

1. Partial derivative with respect to x:
To find the derivative with respect to x, treat y as a constant and differentiate the equation with respect to x. For each term, multiply the coefficient by the power of x and then subtract 1 from the power of x.
∂(3x^2 - 2xy + y^2)/∂x = 6x - 2y = 0. This is the first equation.

2. Partial derivative with respect to y:
To find the derivative with respect to y, treat x as a constant and differentiate the equation with respect to y. For each term, multiply the coefficient by the power of y and then subtract 1 from the power of y.
∂(3x^2 - 2xy + y^2)/∂y = -2x + 2y = 0. This is the second equation.

Now, we have a system of equations:
6x - 2y = 0 ----------(1)
-2x + 2y = 0 ----------(2)

Let's solve these equations simultaneously.

From equation (2), we can rearrange to get:
2y = 2x, which simplifies to y = x.

Substituting this into equation (1):
6x - 2(x) = 0
6x - 2x = 0
4x = 0
x = 0

So, x = 0 is the x-coordinate of the points where the relation has a horizontal or vertical tangent line.

To determine the corresponding y-coordinate, let's use the original equation:
3x^2 - 2xy + y^2 = 24.

Substituting x = 0:
0 - 2(0)y + y^2 = 24
y^2 = 24
y = ±√24
y = ±2√6

Therefore, the points where the relation has a vertical or horizontal tangent line are (0, 2√6) and (0, -2√6).