Shortest Distance from Curve to Origin: How to Minimize x^2+y^2?

In summary, the distance from curve to origin is the shortest distance between a point on a curve and the origin on a coordinate plane. It is calculated using the Pythagorean theorem and is always a positive value. It is different from the length of the curve and is used in various mathematical and scientific applications.
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Homework Statement

Find the shortest distance from [tex]x^{2}-y^{2}=1[/tex] and the origin.

I was gone when my professor taught on this. I just want some hints to get started...

Homework Equations





The Attempt at a Solution

I graphed it and it's pretty easy to see that the answer is one. I just don't know how to prove it...
 
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  • #2
The square of distance is x^2+y^2, so you have to minimize that. Can you express x^2+y^2 in terms of only one variable?
 

FAQ: Shortest Distance from Curve to Origin: How to Minimize x^2+y^2?

What is the definition of "Distance from curve to origin"?

The distance from curve to origin refers to the shortest distance between a point on a curve and the origin (0,0) on a coordinate plane.

How is the distance from curve to origin calculated?

The distance from curve to origin can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the other two sides. In this case, the hypotenuse represents the distance from the curve to the origin, and the other two sides represent the x and y coordinates of the point on the curve.

Can the distance from curve to origin be negative?

No, the distance from curve to origin is always a positive value. This is because distance is a measure of how far apart two points are, and it cannot be negative.

Is the distance from curve to origin the same as the length of the curve?

No, the distance from curve to origin and the length of the curve are two different measurements. The distance from curve to origin is a straight line distance, while the length of the curve takes into account the shape and direction of the curve.

How is the distance from curve to origin used in mathematics?

The distance from curve to origin is used in various mathematical and scientific applications, such as calculating the shortest distance between two points on a graph, finding the equation of a tangent line, and determining the rate of change of a function at a specific point.

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