Find the equation of the tangent line to the curve ( a hyperbola)?

Question

(x^2/36)-(y^2/9)=1 at the point (-8,1/6sqrt(252)). The equation of this tangent line can be written in the form y=mx+b where
m=________
b=________

Answer 1

Using implicit differentiation we find dy/dx as

2x/36 - 2y/9 dy/dx = 0

dy/dx = x/(4y)

x1 = -8 and y1 = sqrt(252)/6

dy/dx = m = -8/[4sqrt(252)/6] = -12/sqrt(252)

Now we just construct the line through this point.

y - y1 = m(x - x1)

y - sqrt(252)/6 = -12/sqrt(252)[x - (-8)]

y - sqrt(252)/6 = -12/sqrt(252)(x + 8)

y = -12/sqrt(252)(x + 8) + sqrt(252)/6

y = -12/sqrt(252)x + 8[-12/sqrt(252)] + sqrt(252)/6

y = -sqrt(252)x/21 - 8sqrt(252)/21 + sqrt(252)/6

y = -sqrt(252)x/21 - 16sqrt(252)/42 + 7sqrt(252)/42

y = -sqrt(252)x/21 - 9sqrt(252)/42

y = -sqrt(252)x/21 - 3sqrt(252)/14

So from this we get

m = -sqrt(252)/21
b = -sqrt(252)/14

Answer 2

m is the 'slope': how fast does y increase when x increases by 1.
b is the intercept (where does the line cross the y axis -- or what is the value of y when x = 0).

The slope of the tangent line will be the instantaneous slope that the hyperbola has at the point of tangency.

This can be found by finding the differential of the curve (dy/dx = some equation) and, in this equation, you put in the values of the point: x= -8 and y = 1/6sqrt(252)

That gives you the m for the tangent line (m = dy/dx since m represents how fast y changes in relation to x).

If you get this far, you now have all values for y = mx + b, except b; therefore it should be easy to find b.
m = dy/dx in solving the differential.
x = -8
y = 1/6sqrt(252)

-----

x^2 / 36 = 1 + y^2 / 9

Differentiate both sides:
D( x^2/36) = D(1 + y^2/9)
[D(x^2)]/36 = D(1) + [D(y^2)]/9
[2x dx]/36 = 0 + [2y dy]/9

rearrange until you get dy/dx on one side and everything else on the other. You will have a 'x' and a 'y' on the othre side, which you will replace with the values at the point:
x = -8 and y = 1/6sqrt(252)

Answer 3

(x^2/36)-(y^2/9)=1
x/18 -2yy'/9 = 0
-2yy'/9 = -x/18
y' =m = x/(4y)
m = -8/(24sqrt(252)) =-8/(24*6sqrt(7)) = -1/(18sqrt(7))
m = -sqrt(7)/126
y = -sqrt(7)x/126 + b
1/(36sqrt(7)) = -8/(sqrt(7))/126 +b
b = 69sqrt(7)/756 = 23sqrt(7)/252