Where does the normal line intersect the second time?

In summary, the normal line is a line that intersects a curve at a specific point and is perpendicular to the curve at that point. It can only intersect the curve twice, and its importance lies in its ability to approximate the curve and determine rates of change and direction of motion. To find the point of intersection for the normal line, one must first find the slope of the tangent line at the first intersection point, take the negative reciprocal, and use it to find the slope of the normal line. Lastly, the point-slope form can be used to solve for the point of intersection.
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Where does the normal line to the ellipse x2-xy+y2=3 at the point (-1,1) intersect the ellipse a second time?So I took the derivative of the equation to get:
y'=(2x-y)/(x-2y)
Then I put (-1,1) into the equation, to get a slope of 1.
So, for the normal line I got a slope of -1, which has the equation y=-x.

And that's as far as I got, I know the answer is suppose to be (1,-1).
 
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Nevermind, I figured it out.
 

Related to Where does the normal line intersect the second time?

1. Where does the normal line intersect the second time?

The normal line will intersect the curve a second time at the point where the derivative of the curve is equal to the negative reciprocal of the slope of the tangent line at the first intersection point.

2. How do I find the point of intersection for the normal line?

To find the point of intersection for the normal line, you need to first find the slope of the tangent line at the first intersection point. Then, take the negative reciprocal of that slope and use it to find the slope of the normal line. Finally, use the point-slope form to find the equation of the normal line and solve for the point of intersection.

3. Why is the normal line important?

The normal line is important because it is perpendicular to the tangent line at the point of intersection. This means that it is the line that best approximates the curve at that point, making it useful for calculating rates of change and finding the direction of motion for objects on the curve.

4. What is the difference between the normal line and the tangent line?

The normal line and the tangent line are both lines that intersect a curve at a specific point. However, the normal line is perpendicular to the curve at that point, while the tangent line is parallel to the curve at that point. In other words, the normal line represents the best linear approximation of the curve at that point, while the tangent line shows the direction of motion for objects on the curve at that point.

5. Can the normal line intersect the curve more than twice?

No, the normal line can only intersect the curve twice. This is because the normal line is always perpendicular to the curve at the point of intersection, and a line can only be perpendicular to a curve at a maximum of two points.

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