Important Facts and Previous Year Questions On Circle

The locus of points which moves in a plane such that its distance from a fixed point is always constant is termed as a circle. The fixed point is termed as the ‘centre’ of the circle and the distance from the centre to a point on the circle is termed as the ‘radius’ of the circle. As far as the JEE exam is concerned, the circle is an important topic in the conic section. Students can expect 1-2 questions from this topic. So JEE aspirants are recommended to gain deep knowledge on this topic.

There are different important formulas related to the circle like standard form, parametric form, general form etc. In this article, we will discuss important equations and previous year questions from circles. This article helps students to have a quick revision on the topic.

Important Formulas

1. Standard Equation:

If the centre is (0, 0) and r is the radius, then the equation of the circle is:

x2 + y2 = r2

2. The equation in centre radius form:

If C denotes the centre given by (h, k) and r is the radius of the circle. Consider another point P(x, y) on the circle, then the equation of the circle is given by 

(x-h)2 + (y – k)2 = r2. 

3. General equation:

If the centre is (-g, -f) and r is the radius, then the equation is:

Here r2 = g2 + f2 – c.

4. Equation of circle through three non-collinear points:

If P(x1, y1), Q(x2, y2) and R(x3, y3) are the non-collinear points, then the equation is given by

5. Equation of a circle with points as extremities of diameter:

If P(x1, y1) and Q(x2, y2) are the extremities, then the equation is given by

(x-x1)(x-x2) + (y-y1)(y-y2) = 0

6. Equation of circle when it touches both axes:

If the centre is (a,a) and radius of circle is a, then the equation is given by

(x-a)2 + (y-a)2 = a2.

7. Equation of tangent:

If circle is given by x2 + y2 + 2gx + 2fy + c = 0 and tangent be at a point P(x1, y1), then equation of tangent to the circle is given by 

Main Points to Remember

1. In a circle, the arcs intercepted by two congruent chords are congruent.
2. Angles in the same segment are equal.
3. The arcs intercepted by two congruent chords in a circle are congruent.
4. If the diameter or the radius of a circle is perpendicular to its chord, then it divides the chord into two equal parts. 

Students are recommended to practise the previous year questions so that they can have an idea about what type of questions are asked from this topic. Let us have a look at an example of  JEE Circle Previous Year Questions With Solutions. The step by step solutions will help students to easily understand the problems.

Example:

Let the tangents drawn from the origin to the circle x2 + y2 -8x – 4y + 16 = 0 touch it at the point A and B. Then (AB)2 is equal to

(a) 32/5

(b) 64/5

(c) 52/5

(d) 56/5

Solution:

Given equation is  x2 + y2 -8x – 4y + 16 = 0.

2gx = -8x

g = -4

2fy = -4y

f = -2

Centre is (4,2).

r2 = g2 + f2 – c

r2 = 16+4-16 = 4

r = 2

Consider triangle OBC

tan θ = 4/2= 2

sin θ = 2/√5

Consider triangle BDC

sin θ = BD/2 

BD = 4/√5

Length of chord = AB = 8/√5

(AB)2 = 64/5

Hence option (b) is the answer.