Find the equation of the tangent line to the hyperbola 3x^2-4y^2=8 which is/are perpendicular to the line 2x-3y+6=0. Give the answers in general form.

the line has slope 2/3

So, the normal needs slope -3/2

3x^2-4y^2=8

6x - 8yy' = 0
y' = 3x/4y

So, we need 3x/4y = -3/2
y = -x/2

3x^2-4(-x/2)^2 = 8
3x^2-x^2 = 8
x = ±2
y = ∓1
So, we want lines through (2,-1) and (-2,1) with slope -3/2

y-1 = -3/2 (x+2)
y+1 = -3/2 (x-2)

See the graphs at

http://www.wolframalpha.com/input/?i=plot+3x^2-4y^2%3D8%2C+2x-3y%2B6%3D0%2C+y+%3D+-3%2F2+%28x%2B2%29%2B1%2Cy+%3D+-3%2F2+%28x-2%29-1

To find the equation of the tangent line to the hyperbola that is perpendicular to the line, we need to follow these steps:

1. Find the derivative of the hyperbola equation.
2. Find the slope of the tangent line by using the derivative.
3. Find the slope of a perpendicular line using the given line equation.
4. Set the two slopes equal to each other and solve for the x-coordinate of the point of tangency.
5. Substitute the x-coordinate into the hyperbola equation to find the corresponding y-coordinate.
6. Substitute the x and y values into the equation of the tangent line.

Let's go through these steps:

1. Find the derivative of the hyperbola equation:
Differentiating both sides with respect to x, we get:
6x - 8y * dy/dx = 0
Simplifying, we have:
dy/dx = 3x / (4y)

2. Find the slope of the tangent line:
Plug in the x-coordinate of the point of tangency into the derivative obtained in step 1 to find the slope of the tangent line.
Let the x-coordinate of the point of tangency be 'a'. Then, the slope of the tangent line is equal to dy/dx when x = a.
So, the slope of the tangent line is: m1 = (3a) / (4y)

3. Find the slope of the perpendicular line:
Given line equation is 2x - 3y + 6 = 0.
Rearrange the equation in slope-intercept form (y = mx + c) to determine its slope.
2x - 3y + 6 = 0
-3y = -2x - 6
y = (2/3)x + 2
The slope of this line (m2) is 2/3.

4. Set the slopes equal to each other:
Since the tangent line is perpendicular to the given line, the product of their slopes should be -1.
So, (3a) / (4y) = -1.
Rewrite it as: 4y = -3a.

5. Find the coordinates of the point of tangency:
Substitute the x-coordinate (a) into the hyperbola equation:
3a^2 - 4y^2 = 8
Substitute 4y = -3a (from step 4):
3a^2 - 4 * (-3a)^2 = 8
Simplify and solve for 'a'.

6. Substitute the x and y values into the equation of the tangent line:
Plug the solution from step 5 into the equation of the tangent line to obtain the full equation in general form.