Points on ellipse where tangent slope=1

[Feodalherren]

Homework Statement

7
Find the points on the ellipse x^2 + 2y^2 = 1 where the tangent line has slope 1

Homework Equations

The Attempt at a Solution

I got the correct X and Y values but this gives me four possibilities and the answer key says there are two points.

I got x = +/- sqrt 2/3 and y = +/- sqrt 1/6
The values are correct but how am I supposed to know which coordinate to pare with which?

For an example, how do I know if it's (-sqrt 2/3, -sqrt 1/6) or (sqrt 2/3, -sqrt 1/6) ?

 

[Sunil Simha]

I got x = +/- sqrt 2/3 and y = +/- sqrt 1/6
The values are correct but how am I supposed to know which coordinate to pare with which?

For an example, how do I know if it's (-sqrt 2/3, -sqrt 1/6) or (sqrt 2/3, -sqrt 1/6) ?

It helps to draw a diagram. If the slope is +1, aren't there only two possible tangents? In which quadrants do their points of intersection with the ellipse lie?

 

[HallsofIvy]

From $x^2+ 2y^2= 1$ you get 2x+ 2yy'= 0 so that y'=- x/y= 1. That gives y= -x and putting that into the equation of the ellipse, $x^2+ 2x^2= 3x^2= 1$ so that $x= \pm\sqrt{3}/3$.

That's what you got , right? And that gives four points on the ellipse?

But your condition is y= -x. Putting $x= \sqrt{3}/3$ and $x= -\sqrt{3}/3$ into that gives only two points.

(The other two points on the ellipse give slope -1.)

 

[Feodalherren]

Oh, DUH! Thanks guys :D.

 

[Mark44]

It helps to draw a diagram.

It's hard to overemphasize the importance of this advice. A halfway decent diagram can give you insight that you can't get just fiddling with equations.

If the slope is +1, aren't there only two possible tangents? In which quadrants do their points of intersection with the ellipse lie?

 

[Feodalherren]

I actually did draw a diagram the first time I did it but I still got four points, it just didn't strike me to draw the tangent lines. If this question is on my test today (in 1.5hrs) I'll nail it. Thanks again.

 

[Mark44]

I actually did draw a diagram the first time I did it but I still got four points, it just didn't strike me to draw the tangent lines.

Try to draw a graph that represents the situation; namely, an ellipse with some points where the slope of the tangent line equals 1. If you got four points, your drawing wasn't an accurate representation of the question.

 

[Feodalherren]

I realize that now. It became painfully obvious as soon as I read Mr Simha's answer.