Question

How do you convert the polar equation $r=5\sec \theta $ into rectangular form?

Answer 1

Hint: We know that the polar coordinates is a way of representing a point in a plane like the rectangular coordinates. An equation that includes the rectangular coordinates is called a rectangular equation. The equation in terms of the polar coordinates is called the polar equation. The polar coordinates and the rectangular coordinates are related to each other. We can use the relations to convert a polar equation to a rectangular equation. We will use $x=r\cos \theta .$

Complete step by step solution:
Let us consider the given polar equation $r=5\sec \theta .$
We are asked to convert the given polar equation into the corresponding rectangular equation.
We know that the rectangular coordinates are used to represent the position of a point in a plane.
Similarly, the polar coordinates are also used to represent the position of a point in a plane.
The equations containing the rectangular coordinates are called rectangular equations and the equation containing the polar coordinates the polar equations.
We know that we can find relations connecting the rectangular coordinates and the polar coordinates.
We can convert the given equation as $r\cos \theta =5.......\left( 1 \right).$
We know that $x=r\cos \theta .......\left( 2 \right)$
When we compare the equations $\left( 1 \right)$ and $\left( 2 \right),$ we will get $x=r\cos \theta= 5.$

Hence the rectangular form of the given polar equation is $x=5.$

Note: We know the trigonometric identity given by $\sec \theta =\dfrac{1}{\cos \theta }.$ We have changed the given equation using this identity as $r=5\sec \theta =\dfrac{5}{\cos \theta }.$ This is changed to $r\cos \theta =5$ There are relations that connect the rectangular and the polar coordinates. They are given by $y=\sin \theta , {{x}^{2}}+{{y}^{2}}={{r}^{2}}.$