With the help of IIT JEE Circle Notes PDF, you can revisit the topics that you studied in Class 9, 10 and 11.

By Mohit ChauhanPublished : July 29, 2020 , 16:02 IST In two dimensional geometry, the circle is considered one of the most widely asked topics in which is definitely asked in JEE Main / JEE Advanced. From this section 2-3 questions are definitely asked in the JEE exam. Download the Circle IIT JEE Notes PDF for your revision.

## IIT JEE Notes for Circle

Here you can revise all the concepts of circles that can be tested in IIT JEE Mains & Advanced exam. Also, you may download IIT JEE Mains Circle Notes PDF here.

### 1. Circle

A circle is the locus of a point whose distance is constant from a given point around which it moves.

### 2. Equation of circle

A circle with center (a, b) and radius r is given as

(x + a)2 + (y + b)2 = r2

### 3. The general equation of the circle

Any second order curve can be defined as

Ax2+Bxy+Cy2+Gx+Hy+D=0

The equation represents a circle on the conditions:

A=C and B=0.

Thus, the general equation of the circle is

Ax2+Ay2+Dx+Ey+G=0

Or

x2+y2+2gx+2hy+c=0

Rewrite the equation of the circle

x2+y2+2gx+2hy+c=0

x2+y2+2gx+2hy+c+g2+h2 = g2+h2

(x + g)2+(y + h)2= g2+h2-c

The center of the circle is (-g,-h) and the radius of the circle is Note: For a real circle  g2+h2-c≥0

### 4.Diametric form of a circle

If the endpoints of a diameter to the circle are (x1,y1) and (x2,y2) ### 5. Position of a point w.r.t circle

If S: x2 + y2 +2gx+2hy+c=0

Let point be P (x1, y1)

Sp: x12 + y12 +2gx1+2hy1 + c=0

If Sp > 0 point lies outside the circle.

If Sp = 0 point lies on the circle.

If Sp < 0 point lies inside the circle.

If the point lies outside the circle, the greatest distance of the point from the circle is

(CP + r) and the least distance of the point from the circle is (CP – r).

### 6. Parametric equation of the circle

The parametric coordinates of the circle are (x1 + rcosθ, y1 +rsinθ )  where the (x1 , y1) is the center of the circle and r is the radius of the circle,  is the angle made by the point from the centre.

The parametric equation is given by ### 7. Line and Circle

Solve the equation of circle by substituting any one variable from the equation of the line. A quadratic equation in any on a variable is formed. Evaluate the determinant of the equation.

If the discriminant, D > 0, the line is a secant.

If the discriminant, D=0, the line is a tangent.

If the discriminant, D<0, then the line doesn’t touch the circle.

The equation of tangent of the circle

At a point (x1, y1 )

Form the equation of tangent by replacing

x2→xx1

y2→yy1

x→(x+x1)/2

y→(y+y1)/2

in the equation of the circle.

### 8. Slope form

If the equation of the tangent is y=mx + c, where m is the slope of the tangent and c is arbitrary constant, then Slope of the tangent to a circle x 2 + y2 = r2 is ### 9. Length of the tangent

If the equation of circle is given as, S: x2 + y2 +2gx+2hy+c=0, the length of the tangent to the point P(x1,y1) is given by

LT = √SP

Where Sp: x12 + y12 +2gx1+2hy1 + c

### 10. The equation of chord of a circle when the median of the chord is given.

If the median of the chord is (h, k), then the equation of chord to the circle

S: (x + a)2 + (y + b)2 = r2

is

xh+yk-r2=h2+k2

or

(T=SP)

### 11. The equation of chord of contact

The equation of chord from the contact  to the circle S: (x + a)2 + (y + b)2 = r2 is

xh+yk-r2=0

### 12. The equation of pair of tangents

If the circle is given as S: x2 + y2 = r2

Sp: x12 + y12 = r2

TP: xx1+yy1 = r2

SSP=TP2

### 13. The family of circles passing the points of intersection of two circles

If the circle S’ and S’’ are intersecting, the family of circles that can be formed from the point of contacts S’ and S’’ is S’ + λS’’=0

### 14. The family of circles passing the points of intersection of a circle and a line

If the circle S and line L are intersecting, the family of circles that can be formed from the point of contacts S and L is S+λL

### 14.1 Common tangent to two circles

There are two types of common tangents to two circles,

(a) Internal common tangent

(b) External common tangent ### 15. The position of circles and the number of common tangents

Let C1 and C2 denote the center of the circles, r1 & r2 be the radii of the circles and C1C2  be the distance between the centers.

(a) If C1C2 > r1 + r2

There are 4 common tangents (b) If C1C2 = r1 + r2 There are 3 common tangents.

(c) If |r1 - r2 |< C1C2 < r1 + r2 There are 2 common tangents.

(d) If C1C2 = |r1 - r2| There is one common tangent.

(e) If C1C2 <|r1 - r2| There is no common tangent.

### (a) External tangent ### (b) Internal tangent ### 17. The condition of orthogonality of two circles

Two circles intersect each other at 90o when

2g1g2 + 2f1f2 = c1+c2

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