Equations Of Circles

A circle is a two-dimensional figure made up of points next to one another and distanced equally from a fixed point. The centre of the circle is the fixed point in this curved plane figure, while the radius is the common distance between the points from the centre. A diameter is a line that crosses from the centre of the circle, starting from one point to the other. The inside of a circle, the exterior of a circle, and the three primary sections of a circle are on the circle. However, before going to the equation of circles, it is important to know a few terms that will help solve the equation of circles questions.

Forms of Equation of Circle

The formulas for calculating the area and circumference of a circle are not the same as the equation for a circle. This equation is employed in a variety of circle issues in coordinate geometry. The equation of the circle is required to represent a circle on the Cartesian plane. If we know the centre and radius of a circle, we can draw it on paper. Similarly, we can draw a circle on a Cartesian plane if we know the coordinates of the centre and radius. A circle can be depicted in a variety of ways:

The common form of an equation:

x² + y² + 2gx + 2fy + c = 0 is the general form of the equation of the circle. Where g, f, and c are constant. This general form is used to obtain the circle’s centre and radius coordinates. In contrast to the conventional form, this equation of a circle makes it difficult to find any related properties about any given circle. So, to quickly convert from the general form to the standard form, we’ll use the completing the square formula.

The standard form of an equation:

The standard equation of a circle provides precise information about the circle’s centre and radius, making it much easier to read the circle’s centre and radius at a glance. A circle’s usual equation with its centre at (x1y1) and radius r is (x – x1)² + (y – y1)²  = r², where (x1y) is at a random point on the circumference of the circle.

Parametric equation of a circle:

We know that x² + y²+ 2hx + 2ky + C = 0 is the general form of the circle equation. Let’s imagine we start at a general position on the circle’s perimeter (x, y). The angle formed by the line connecting this general point and the circle’s centre (-h, -k) equals θ. The circle’s parametric equation is x² + y² + 2hx + 2ky + C = 0, where x = -h + rcos θ and y = -k + rsinθ.

Polar equation of a circle:

The polar form of the circle equation is nearly identical to the parametric form of the circle equation. For a circle centred at the origin, we commonly write the polar version of the equation of a circle.

Steps To Find The Equation Of A Circle

Depending on where the circle’s centre is in the Cartesian plane, we’ve seen different ways to represent the equation of a circle. When the dimensions of the centre are inferred, the following steps can be used to write the equation of a circle:

Step 1: Determine the radius of the circle and the coordinates of the circle’s centre (x1, y1).

Step 2: Determine the radius and centre of the circle using the circle formula (x – x1)² + (y – y1)²= r².

Step 3: The equation of the circle will be found after simplifying the equation.

Equation of Circle

(I) Using the equation of the circle formula, find the centre and radius of the circle whose equation is (x – 1)2 + (y + 2)2 = 9.

Solution:

We will use the circle equation to determine the centre and radius of the circle.

The first step is to compare

(x−1)²+(y+2)² = 9(x−1)²+(y+2)²=9 with

(x−x1)²+(y−y1)² =r²(x−x1)²+(y−y1)²=r²,

 We get

x1 = 1, y1 = -2 and r = 3

So, the centre and radius are 1, -2, and 3, respectively.

Answer: The centre of the circle is (1, -2), and its radius is 3.

(II) Find the equation of a circle where the centre passes through the origin.

Solution:

We know, the equation of a circle is given by,

(x – x1)² + (y – y1)² = r²,

where x1 and y1 are the coordinates of the circle’s centre; and

r is the radius.

Since the circle passes through the origin, x1 and y1 will be zero.

Answer: The standard equation of the circle passing through the origin will be x² + y² = r²

Conclusion

A circle is a two-dimensional shape created by forming a curve from the centre that is the same distance all the way around. Circumference, radius, diameter, sector areas, chords, tangents, and semicircles are all parts of a circle. Only a handful of problems on circles questions require straight lines, so it’s important to be familiar with the formulas and units of measurement. A circle is a form in which all of its points are at the same distance from the centre. The centre of a circle is given its name. Because its centre is at point A, the circle to the right is named circle A. A wheel, a dinner plate, and a coin are all instances of circles in the real world.