Equation of the Normal and Ellipses Questions

[ardentmed]

Hey guys,

I have a couple of questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help.

Question:

For 1a, I used implicit differentiation and isolated dy/dx.

This gave me the following answer:

dy/dx = (-3x^2 -4y)/(4x+12y)

Which I then used for 1b and substituted (1,1) for x and y respectively to compute:

y-1= (-7/16) * (x-1)

As for 2, the normal is perpindicular to the tangent line and is thus the negative reciprocal. Thus, if the tangent line's slope is 1, the normal is -1/1= -1.

Thus, I used that for the equation of the line through (-1,1) to get:

(1,-1) (and the other point was already given by the question)

This was obtained by using y=-x as the equation and then substituting x=-y into the x^2 + y^2 = 1 elipse expression to get x^2=1. Thanks in advance.

 

[anemone]

For 1a, I used implicit differentiation and isolated dy/dx.

This gave me the following answer:

dy/dx = (-3x^2 -4y)/(4x+12y)

Which I then used for 1b and substituted (1,1) for x and y respectively to compute:

y-1= (-7/16) * (x-1)

Correct for both a and b part.

As for 2, the normal is perpindicular to the tangent line and is thus the negative reciprocal. Thus, if the tangent line's slope is 1, the normal is -1/1= -1.

Thus, I used that for the equation of the line through (-1,1) to get:

(1,-1) (and the other point was already given by the question)

This was obtained by using y=-x as the equation and then substituting x=-y into the x^2 + y^2 = 1 elipse expression to get x^2=1. Thanks in advance.

You've figured out everything correctly so far and now, since solving for $x^2=1$ gives $x=\pm1$, you've to answer the last part of the problem 2 by determining at which point the normal line at $(-1,\,1)$ intersects the ellipse the second time. Can you see how proceed?

 

[ardentmed]

Correct for both a and b part.
You've figured out everything correctly so far and now, since solving for $x^2=1$ gives $x=\pm1$, you've to answer the last part of the problem 2 by determining at which point the normal line at $(-1,\,1)$ intersects the ellipse the second time. Can you see how proceed?

No, I don't quite get it. Isn't the final answer just (1,-1)?
Thanks in advance.