I need to find the equations of the tangent and normal line for x^2-y^2=16, at the point (-5,3). I have to show every detail for my work...then use graphing software to enter my equations, so can anyone help me showing all the details of the work so that I can figure it out?? Thanks, a million-trillion!!
Of course, I'd be happy to assist you with finding the equations of the tangent and normal lines for the given equation at the point (-5,3). Here are the step-by-step instructions:
1. Start with the given equation: x^2 - y^2 = 16.
2. Differentiate both sides of the equation with respect to x to find the derivative: d/dx (x^2 - y^2) = d/dx (16).
Using the chain rule, we get:
2x - 2y * (dy/dx) = 0
3. Solve the derivative equation for dy/dx to find the slope of the tangent line:
dy/dx = (2x) / (2y)
4. Plug in the coordinates of the given point (-5, 3) into the derivative to find the slope at that point:
dy/dx = (2 * (-5)) / (2 * 3) = -10/6 = -5/3
5. Now we have the slope of the tangent line. To find the equation of the tangent line, we'll use the point-slope form:
y - y1 = m(x - x1),
where (x1, y1) is the given point and m is the slope.
Substitute the values into the equation:
y - 3 = (-5/3)(x - (-5))
Simplify the equation:
y - 3 = (-5/3)(x + 5)
Expand and rearrange the equation:
y = (-5/3)x - 25/3 + 9/3
y = (-5/3)x - 16/3
Therefore, the equation of the tangent line is y = (-5/3)x - 16/3.
6. To find the equation of the normal line, we can use the fact that the slope of the normal line is the negative reciprocal of the slope of the tangent line. So, the slope of the normal line is 3/5.
Using the point-slope form again:
y - 3 = (3/5)(x - (-5))
Simplifying the equation:
y - 3 = (3/5)(x + 5)
y - 3 = (3/5)x + 15/5
Rearranging the equation:
y = (3/5)x + 15/5 + 3/5
y = (3/5)x + 18/5
Therefore, the equation of the normal line is y = (3/5)x + 18/5.
To verify the results and visualize the lines on a graph, you can use any graphing software by entering the equations of the tangent and normal lines:
- Tangent line: y = (-5/3)x - 16/3
- Normal line: y = (3/5)x + 18/5
I hope this explanation helps you understand the process. Good luck with your work!