# Equation of a Plane Curve

## Equation of a Plane Curve in Cartesian Coordinates

Let a Cartesian coordinate system and some curve *C* be given on a plane. Consider an equation relating two variables *x* and *y*:

The above equation is called the implicit equation of the curve *C*, if the coordinates (*x*, *y*) of any point lying on the curve satisfy this equation and, respectively, the coordinates (*x*, *y*) of any point not lying on the curve do not satisfy this equation.

So for example the equation of a circle of radius *R* centered at the point *M*(*a*, *b*) has the form

If an equation of a curve can be solved for one of the variables *y* or *x*, then we get an explicit form of the equation:

## Parametric Representation of a Curve

It is often convenient to express the coordinates of the points (*x* ,*y*) of a curve *C* using the third auxiliary variable *t* (called a parameter):

where the functions *φ*(*t*) and *ψ*(*t*) are assumed to be continuous in some range of the parameter *t*.

Note that by eliminating the parameter *t* from both these equations we can find a relationship between *x* and *y*, that is the general equation of the curve in the form

If the variable *t* is the time counted from some initial moment, then the functions *φ*(*t*) and *ψ*(*t*) determine the law of motion of the body.

As an example, we write the parametric equations of an ellipse:

The ellipse equation in parametric form can be easily converted to standard form:

## Equation of a Curve in Polar Coordinates

The form of the equation of a curve *C* depends not only on the curve itself, but also on the choice of the coordinate system.

If a curve in the Cartesian coordinate system is described by the equation *f*(*x*, *y*) = 0, then in order to obtain the equation of the same curve with respect to any other coordinate system, it is enough to substitute instead of *x* and *y* their expressions in terms of the new coordinates. In case of polar coordinates, the variables *x* and *y* are expressed in terms of polar coordinates *ρ* and *φ* as follows:

Substituting these relations into the equation of the curve in Cartesian coordinates, we obtain its equation in polar coordinates in the form

Many curves have a simpler equation in polar coordinates than in Cartesian coordinates. Let's take a cardioid as an example.

The cardioid shown in the figure is described by a simple polar equation

The equation describing this curve in Cartesian coordinates has a more complex form:

## Solved Problems

### Example 1.

A point *M* moves along a curve so that it is equidistant from points *A*(3,0) and *B*(-3,6). Find the equation of the curve.

Solution.

Let the coordinates of the point M be (*x*,*y*). Since

we get

Simplify this equation:

We got the equation of a straight line.

### Example 2.

A point *M* moves along a curve so that it is equidistant from the origin and point *B*(-4,2). Write the equation of the curve.

Solution.

Assuming that *M*(*x*,*y*) is an arbitrary point of the curve, we can write:

Substituting the known coordinates, we get

or

It can be seen that this curve is a straight line.

### Example 3.

Find the locus of points equidistant from the *y*-axis and point *B*(2,0).

Solution.

Let *M*(*x*,*y*) be an arbitrary point of the given set.

By the condition of the problem,

Therefore

Solve the equation for *x*:

We obtained a parabola equation. Its graph is rotated \(90^\circ\) clockwise from the standard position and opens to the right.

### Example 4.

Points *A* and *B* are given on a plane. Find the locus of points *M* that are twice as distant from *A* than from *B*.

Solution.

Let the origin be at point *A* and the *x*-axis pass through point *B*. Suppose point *B* has coordinates (*b*,0).

It follows from the condition that

Using the distance formula between two points, we have

Simplify this expression:

Complete the square for the terms with *x*:

or

We see that the locus of points is a circle of radius \({\frac{2b}{3}}\) centered at \(\left({\frac{4b}{3}, 0}\right).\)