Posted by Anonymous on Wednesday, October 7, 2009 at 3:42pm.
I will give you the steps.
1. find dy/dx for the hyperbola
(you should get dy/dx = 3x/y)
2. if 2x + 9y = k is a normal, then its slope is -2/9
so the slope of the tangent at the point of contact should be 9/2
3. set dy/dx = 9/2 and solve for x
4 sub in that x into the original hyperbola and solve
( I got y = ±2)
5. find the x's for each of those two y's.
6. You now have the points of contact of the tangents, and the points of contact of the normals.
7 sub in the points from #5 into the original normal equation to find k
There are of course two of those.
Alright, this one involves a fair bit of complicated calculus, so I'm assuming you have the basics down:
First, some definitions. A normal line is a line that is perpendicular to another function at a given point. That means that the slope of the normal line is the negative reciprocal of the derivate at a given point.
We can find the slope of the given normal line (2x+9y+k=0): slope of the normal is -2/9
This means that the derivative of 3x^(2)-y^(2)=23 at some point is equal to the negative reciprocal of (-2/9), or (9/2). Let us first find the general derivative of 3x^(2)-y^(2)=23. By doing implicit differentiation, we get dy/dx=3x/y .
So we know that 3x/y=9/2. If we do some cross multiplication, it follows that y=(2/3)x. Now we can actually substitute for why into our polynomial and solve for x. By substitute I mean do this: 3x^(2)-((2/3)x)^(2)=23 [substituting (2/3)x in for y]. When we solve this for x, we get: x=+3, -3. Because y=(2/3)x, y=+2,-2.
So now we have the two points where the derivative of our polynomial has a slope of (9/2): points (3, 2) and (-3,-2)
Now that we have these points, plug the values back into 2x+9y+k=0:
2(3)+9(2)+k=0, k=-24
2(-3)+9(-2)+k=0, k=24
So k=+24,-24
Awesome, thanks for the help!
Related Questions
Algebra 1 - Write an equation of the line containing the given point and ...
Algebra - Write the equation of the line containing the given point and parallel...
algrbra 1 - Write an equation of the line containing the given point and ...
calculus - write equation of line that passes through the point 1)parallel to ...
alegbra - Solve by the subsitution method. 2x+4y=-6 -3x+y=23 y=23+3x 2x +4(23+3x...
Math - I had an assignment containing 45 problems, and I am unsure of 4, can you...
MATH - I had an assignment containing 45 problems, and I am unsure of 4, can you...
Calculus - The function f(x) is defined as f(x) = -2(x+2)(x-1)^2 = -2(x^3 -3x + ...
Math - Can someone please check my work for me? Write an equation of the line ...
Algebra - Hello, can someone tell me if I got the following solve and check work...
For Further Reading
