IIT-JEE Main Notes on Circle

M AT H E M AT I C S
STUDY MATERIAL

CIRCLE

IIT-JEE

, CONTENTS

CIRCLE
1. Theory


2. Solved Problems


(i) Subjective Problems


(ii) Single Choice Problems

Multiple Choice Problems


(iii) Comprehension Type Problems

Matching Type Problems

Assertion-Reason Type Problems


3. Assignmentsi


(i) Subjective Questions


(ii) Single Choice Questions


(iii) Multiple Choice Questions


(iv) Comprehension Type Questions

Matching Type Questions

Assertion-Reason Type Questions


(v) Problems Asked in IIT-JEE


4. Answers

, CIRCLE
IIT- JEE SYLLABUS
Equation of a circle in various forms, Equations of tangent, Normal and chord, Parametric equations of
a circle, Intersection of a circle with a straight line or a circle, Equation of a circle through the points of
intersection of two circles and those of a circle and a straight line, Locus Problems.




CONTENTS
INTRODUCTION
♦ Definition of a circle
♦ Intercept made on axes
Many objects in daily life, which are round in
♦ Position of a point w.r.t a circle shape like rings, bangles, wheels of a vehicle
♦ Position of a line w.r.t a circle etc. are the examples of a circle. In terms of
♦ Definition of tangent and normal mathematics, circle is an important locus of a
point in two dimensional coordinate geometry.
♦ Definition of chord of contact
Some times circle is also called as a part of conic
♦ Equation of the chord with a given mid-point
section. In this chapter we deal circle as an
♦ Director circle independent topic.
♦ Equation of a chord of a circle
♦ Equation of the lines joining the origin to
the points of intersection of a circle and a
line
♦ Use of the parametric equations of a
straight line intersecting a circle
♦ Radical axis
♦ Family of circles
♦ Orthogonal circles
♦ Common chord of two intersecting circles
♦ Position of a circle w.r.t a circle and
common tangents
♦ Locus problems

1
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, 1. DEFINITION OF CIRCLE
A circle is the locus of a point which moves in such a way that its distance from a fixed point is constant.
The fixed point is called the centre of the circle and the constant distance, the radius of the circle.

2. EQUATION OF THE CIRCLE IN VARIOUS FORMS
(A) Centre - Radius Form
To find the equation of circle when the centre and radius are given.
Let r be the radius and C (α, β) the centre and P any point on the circle whose coordinates are (x, y).




Then CP2 = r2
i.e. (x – α)2 + (y – β)2 = r2
this equation reduces to x2 + y2 – 2αx – 2βy + α2 + β2 – r2 = 0 ...(i)
nd
General equation of 2 degree :
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 ...(ii)
Let this equation represents a circle with centre (α, β) and radius r.
Hence equation (i) & (ii) represent same circle
Comparing (i) & (ii) we get,

a=b ⇒ coefficient of x 2 = coefficient of y 2 

h=0 ⇒ coefficient of xy = 0 
These are required condition, for general equation of 2nd degree represents a circle
For general equation of a circle put a = b = 1, h = 0 in equation (ii)
Hence general equation of circle is x2 + y2 + 2gx + 2fy + c = 0 ...(iii)
Centre (α, β) = (–g, –f)
∴ c = g2 + f2 – r2 ∴ r2 = g2 + f2 – c

∴ Radius = g2 + f 2 − c


Point circle:
The equation (x – a)2 + (y – b)2 =0 represents a point circle since its radius is zero. So the circle converts
into just a point.

REMARK
(i) g2 + f2 – c > 0 ⇒ equation represents a real circle
(ii) g2 + f2 – c = 0 ⇒ equation represents a point circle
(iii) g +f –c<0 ⇒
2 2
equation represents an imaginary circle


2
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