Relation Between a Circle and a Line in its Plane

For understanding the relation between a coplanar Circle and a Line, let us first have some knowledge about a line, circle and plane.

Circle

A circle is a plane figure bounded by a single curved line, with all straight lines drawn from any point within it to the bounding line being equal. The circumference is the boundary line, and the point is the centre. A circle is a closed two-dimensional shape where all points in the plane are equidistant from a single point known as the “centre.” The line of reflection symmetry is formed by every line that travels through the circle. It also possesses rotational symmetry for every angle around the centre.

Equation of a Circle

The typical form of the equation or formula of a circle is-

(x-h)² + (y-k)² = r²

where (x,y) are the coordinates

(h,k) represents the coordinate of the centre of a circle

and r is said to be the radius of that circle.

It can also be represented as-

x² + y² + 2gx + 2fy + c = 0           ..(1)

 for all values of g, f and c.

 on adding g2 + f2 on both sides of the equation we get,

⇒ g² + f² − c = x² + 2gx + g²+ y² + 2fy + f²

⇒ g² + f²−c = (x+g)²+ (y+f)²           …(2)

on comparing the equation 1 and 2 we observe,

h= −g,

k= −f

a² = g²+ f²−c

Therefore, if the equation of the circle is x² + y² + 2gx + 2fy + c = 0 then center is (−g,−f) and radius is a² = g²+ f²−c.

On comparing the (g²+ f²) and c there can be three possibilities,

1. The radius of the circle is genuine when (g²+ f²) > c.

2. The radius of the circle is zero when (g²+ f²) = c, indicating that the circle is a point that corresponds with the centre. A point circle is one such sort of circle.

3. When (g2+ f2) = c, the circle’s radius becomes imaginary. It is thus a circle with a real centre and an imagined radius.

Example: Say point (2,5) is the center of the circle and radius is equal to 9 cm. Then the equation of this circle will be:

(x – 2)² + (y – 5)² = 92

(x²  +4 −4x) + (y²  + 25 − 10y) = 81

x² + y² − 4x − 10y – 52= 0

Center is Origin-Equation of a Circle

We know that the distance between the point (x, y) and origin (0,0)can be found using the distance formula which is equal to-

√[x2+ y2]= r

Therefore, the equation of a circle, with the center as the origin is,

x²+y²= r²

Where r = radius of the circle.

Polar Equation of a Circle

To find the polar form of the equation of a circle, replace the value of x = r cos θ and y = r sin θ, in x² + y² = a².

Hence,  we get;

(r cos θ)2 + (r sin θ)2 = a2

r² cos2 θ + r2 sin2 θ = a2

r² (cos2 θ + sin2 θ) = a2

as we know that (cos2 θ + sin2 θ)=1

r2  = a2

r = a

This represents the polar equation with radius “a” and center at the origin (0,0) of a circle.

Line-Definition

A  line is a straight 1D figure, or it is a collection of points that does not have a thickness, and it extends endlessly in both directions.

When a line and a circle intersect

A line and a circle can be related in three ways: the line cuts the circle in two places, the line is tangent to the circle, or the line misses the circle. They are-

• There will be two points of contact if the line cuts through the circle.

• There will be one point of contact if the line is tangent to the circle.

• There will be no point of contact if the line misses the circle.

Therefore, when a circle and a line are coplanar, they can have either 0, 1 or 2 points of intersection.

Conclusion

When a circle and a line share the same plane, there are three possibilities: first, if the line misses the circle, second, if the line is one of the tangents of the circle, or third, if the line cuts through the circle, there may be two points of contact.