find the equation of the tangents through the point (8,-1) to the ellipse 2x^2+5y^2=70

Question

mortonsalt · Answer

What do you know about the tangent line of a function?

anonymous · Answer

$$\frac{ xx _{i} }{ a^2 }+\frac{ yy _{i} }{ b^2 }=1$$

mortonsalt · Answer

Basically, the tangent line is the derivative of the function, yeah?

anonymous · Answer

the slope yea

mortonsalt · Answer

If you have slope and a point, you can get the equation by
$$y-y_1 = m(x-x_1)$$

anonymous · Answer

no y=mx+b

mortonsalt · Answer

No, actually, look at it this way instead:
Take the derivative of the function.
$$4x(dx) +10y(dx)=0$$

mortonsalt · Answer

sorry 10y(dy)

mortonsalt · Answer

Plug in your values for x and y at the point (8,-1).

solomonzelman · Answer

I don't understand what you did, excuse me...

mortonsalt · Answer

This is the relation between increments of the variables along the tangent line.

solomonzelman · Answer

Your function is:

$\large\color{black}{ \displaystyle y=\pm\sqrt{-\frac{2}{5}x^2+14 }}$

And since the point (8,-1) is above the x-axis, you choose the +, so just:

$\large\color{black}{ \displaystyle y=\sqrt{-\frac{2}{5}x^2+14 }}$

mortonsalt · Answer

I feel like there's an easier way to do it, though.

solomonzelman · Answer

YOu need the slope of the function at a point x=8, so find f$'$(8), and this the slope of your line. The point you already know.

mortonsalt · Answer

Since the increments are measured from (8, -1), then x-8 and y-(-1).

mortonsalt · Answer

From this, the equation of the line would be
$$32(x-8)-10(y+1)=0$$

solomonzelman · Answer

Basically:

$\large\color{black}{ \displaystyle t(x)=h'(8)\times(x-8)+f(8)}$

(tangent line to the curve at some point is same as Linearization of a curve at some point)

And that is:   $\large\color{black}{ \displaystyle t(x)=h'(8)\times(x-8)+(-1)}$

solomonzelman · Answer

by t(x) I am denoting the tangent line.

mortonsalt · Answer

Don't you think my method would work? I feel like it's a lot simpler.

solomonzelman · Answer

I don't know, but what I did is a typical method.

solomonzelman · Answer

If your line ends to be the same the you are corret as well.

anonymous · Answer

so which is correct way?

mortonsalt · Answer

You can try either way. Do a check to see if the values correspond, then whichever one does, should be the correct way. :)

He did it in the typical way, so maybe you should do his.