Answer:
[tex]y=x-2[/tex]
Step-by-step explanation:
So we are given the formula for the slope of a hyperbola in this form:
[tex]\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/tex].
That formula for the slope is [tex]m=\frac{b^2x}{a^2y}[/tex]
If we compare the following two equations, we will be able to find [tex]a^2[/tex] and [tex]b^2[/tex]:
[tex]\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/tex]
[tex]\frac{x^2}{8}-\frac{y^2}{4}=1[/tex]
We see that [tex]a^2=8[/tex] while [tex]b^2=4[/tex].
So the slope at [tex](x,y)=(4,2)[/tex] is:
[tex]m=\frac{b^2x}{a^2y}=\frac{4(4)}{8(2)}=\frac{16}{16}=1[/tex].
Recall: Slope-intercept form of a linear equation is [tex]y=mx+b[/tex].
We just found [tex]m=1[/tex]. Let's plug that in.
[tex]y=1x+b[/tex]
[tex]y=x+b[/tex]
To find [tex]b[/tex], the [tex]y[/tex]-intercept, we will need to use a point on our tangent line. We know that it is going through [tex](4,2)[/tex].
Let's enter this point in to find [tex]b[/tex].
[tex]2=4+b[/tex]
Subtract 4 on both sides:
[tex]2-4=b[/tex]
Simplify:
[tex]-2=b[/tex]
The equation for the tangent line at [tex](4,2)[/tex] on the given equation is:
[tex]y=x-2[/tex]
Answer: y = x - 2
Step-by-step explanation:
First you take the derivative of each term. d/dx(x²/8) - d/dx(y²/4) = d/dx(1)
x/4 - (y/2)dy/dx = 0
Then you solve for dy/dx: dy/dx = x/2y
Plug in the values: dy/dx = 1
To find the y-intercept, plug in values for y = mx+ b. 2 = 4 + b, b = -2
The equation is y = x - 2
