# Proving Orthogonal Curves: y1=-.5x^2+k & y2=lnx+c

• minase
In summary, the student is requesting help with a problem that asks for two curves to be perpendicular at every point of intersection.
minase
We were given a graded assigment and one of the question asks.
Prove that all curves in the family
y1=-.5x^2 + k (k any constant) are perpendicular to all curves in the family
y2=lnx+ c (c any constant) at their points of intersection.

I found the derivatives of y1 and y2 and they are reciprocal to each other. But it is asking for points of intersection of the curves. Can curves be perpendicular to each other. The graded assigment is due tomorrow if possible if you give an explanation right about now it would be really great.

Yes, curves can be orthogonal to one another, and you've already determined that these two curves are in fact orthogonal to one another because their respective derivatives are lines with negative reciprocal slopes. You can determine the points of intersection of the curves by setting them equal to one another. Don't set both constants = 0 though, since lnx will never intersect with -.5x^2 unless at least one of them has a vertical shift.

Are you asking if it is possible for curves to be perpendicular to one another? Of course it is! For example the circle with center at (1, 0) and radius 1 and the circle with center at (0,1) and radius 1 are perpendicular to one another. Their tangent lines at (0,0) are perpendicular.

You are dealing with 2 families of curves which means that there is a curve of each family passing through every point in the plane. Every point is a "point of intersection" so you don't need to worry about that. Since each curve is given as a function: y(x)= ..., the derivative only depends on x so as long as the derivatives are correct for each x, the families are perpendicular.

By the way, you say "I found the derivatives of y1 and y2 and they are reciprocal to each other." For two curves to be perpendicular, the slopes of their tangent lines must be negative reciprocals. Is that what you meant?

I have the same exact problem. Can someone please show me how to solve
-.5x^2=lnx. Thanks.

If you have "the exact same problem", then you do NOT need to solve that equation. The problem asks you to show that the curves are perpendicular at every point of intersection, NOT to find the points of intersection of two curves with the same k. The first equation gives a family of parabolas so that there is one curve of that family through every point in the plane. The second equation, x must be positive, gives a family of curves through every point in the half plane, x> 0. Every point in the right half-plane, x>0, is a point of intersection!

Can you still show me how to do it .Please. Thanks.

## What are orthogonal curves?

Orthogonal curves are two curves that intersect at a right angle. This means that the tangent lines at the point of intersection are perpendicular to each other.

## What is the process of proving orthogonality between two curves?

In order to prove orthogonality between two curves, we need to show that the slopes of the tangent lines at the point of intersection are negative reciprocals of each other. This means that when multiplied together, they result in a slope of -1.

## Can orthogonal curves have different equations?

Yes, orthogonal curves do not necessarily have to have the same equation. In fact, they often have different equations, but their slopes at the point of intersection are still negative reciprocals of each other.

## How do we prove orthogonality between y1=-.5x^2+k & y2=lnx+c?

To prove orthogonality between these two curves, we first need to find their respective derivatives. Then, we can plug in the x-value of the point of intersection into both derivatives to find the slopes of the tangent lines. If the product of these slopes is -1, then we have proven that the curves are orthogonal.

## What is the significance of proving orthogonality between two curves?

Proving orthogonality between two curves is important because it allows us to understand the relationship between them and how they behave at the point of intersection. It also helps us to solve problems involving these curves, such as finding the area between them or finding the length of their intersection.

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