# Step-By-Step to Find the Shortest Distance Between a Point and a Curve

Finding the shortest distance between a point and a curve is a concept used in critical physics applications such as projectile motion, celestial mechanics, aeronautics, and even when studying ocean currents. With the right mathematical knowledge, computing this distance can be simple and straightforward. This article looks at the basics of the problem and provides a step-by-step solution to help you solve the problem in a more efficient way.

## Definition And Principles

To find the shortest distance between a point and a curve, you need to know its mathematical definition and the relevant principles beforehand. To make it simple, let's assume that a curve is defined by a set of points in a two-dimensional space.

In this case, the shortest distance between the point and the curve is the minimum value of the distance between the point and all the points that make up the curve. This distance is also known as the shortest perpendicular distance.

## Step-By-Step Solution

When you have the equation of the curve available and the coordinates of the point, the solution can be divided into few steps:

1. Find the equation of the line that passes through the point and is perpendicular to the curve.
2. Set the equation of the line equal to the equation of the curve to get an expression for x in terms of y.
3. Substitute this expression for x into the equation of the curve to get an equation in terms of y.
4. Solve the new equation for y.
5. Substitute the value of y back into the equation of the line to get x.
6. Calculate the length of the line segment connecting the point and the curve.

## FAQ:

### How Do I Find The Equation Of The Line?

The equation of the line that passes through a point and is perpendicular to a curve is given by the equation `y-y1=m(x-x1)`, where `(x1,y1)` is the coordinates of the point and m is the slope of the line. The slope of the line can be calculated using the equation `-1/m=-(A/B)`, where A and B are the coefficients of x and y from the equation of the curve respectively.

### How Do I Find The Equation Of The Curve?

An equation of a curve can be found by plotting explicit equations for x and y. Alternatively, implicit equations can also be used depending on the type of the curve.

### How Do I Find The Length Of The Line Segment?

The length of the line segment connecting the point and the curve can be found using the Pythagorean theorem. This theorem states that for any right triangle, the square of the hypotenuse (the line segment) is equal to the sum of the squares of the other two sides. So the length of the hypotenuse can be calculated as `√(x2−x1)2+(y2−y1)2`.

### What Are Implicit Equations?

Implicit equations are equations of the form `F(x,y)=c` where F and c are some constants. These equations can be used to represent a curve and they are usually used to represent curves with complex equations and those which can't be expressed as explicit equations.

### What Are Explicit Equations?

Explicit equations are equations of the form `y=f(x)` where f is some function. These equations can be used to represent any type of curve and are usually used to represent simple curves like parabolas and circles.

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