# IIT JEE Circle Notes, Download PDF

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

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} = r^{2}**

**3. The general equation of the circle**

Any second order curve can be defined as

Ax^{2}+Bxy+Cy^{2}+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

Ax^{2}+Ay^{2}+Dx+Ey+G=0

Or

**x ^{2}+y^{2}+2gx+2hy+c=0**

Rewrite the equation of the circle

x^{2}+y^{2}+2gx+2hy+c=0

x^{2}+y^{2}+2gx+2hy+c+g^{2}+h^{2} = g^{2}+h^{2}

(x + g)^{2}+(y + h)^{2}= g^{2}+h^{2}-c

The center of the circle is (-g,-h) and the radius of the circle is

**Note: For a real circle ** g^{2}+h^{2}-c≥0

**4.Diametric form of a circle**

If the endpoints of a diameter to the circle are (x_{1},y_{1}) and (x_{2},y_{2})

**5. Position of a point w.r.t circle**

If S: x^{2} + y^{2} +2gx+2hy+c=0

Let point be P (x_{1}, y_{1})

S_{p}: x_{1}^{2} + y_{1}^{2} +2gx_{1}+2hy_{1 }+ c=0

If S_{p} > 0 point lies outside the circle.

If S_{p} = 0 point lies on the circle.

If S_{p} < 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 (x_{1} + rcosθ, y_{1} +rsinθ ) where the (x_{1} , y_{1}) 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 (x _{1}, y_{1} ) **

Form the equation of tangent by replacing

x^{2}→xx_{1 }

y^{2}→yy_{1}

x→(x+x_{1})/2

y→(y+y_{1})/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} + y^{2} = r^{2} is

**9. ****Length of the tangent**

If the equation of circle is given as, **S: x ^{2} + y^{2} +2gx+2hy+c=0, **the length of the tangent to the point P(x

_{1},y

_{1}) is given by

L_{T} = √S_{P}

Where S_{p}: x_{1}^{2} + y_{1}^{2} +2gx_{1}+2hy_{1 }+ 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} = r^{2}

is

xh+yk-r^{2}=h^{2}+k^{2}

**or**

**(T=S _{P})**

**11. The equation of chord of contact**

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

xh+yk-r^{2}=0

**12. The equation of pair of tangents**

If the circle is given as S: x^{2} + y^{2} = r^{2}

S_{p}: x_{1}^{2} + y_{1}^{2} = r^{2}

T_{P}: xx_{1}+yy_{1} = r^{2}

SS_{P}=T_{P}^{2}

**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 C_{1} and C_{2} denote the center of the circles, r_{1} & r_{2} be the radii of the circles and C_{1}C_{2 } be the distance between the centers.

**(a) If C _{1}C_{2 }> r_{1} + r_{2}**

There are 4 common tangents

**(b) If C _{1}C_{2 }= r_{1} + r_{2}**

There are 3 common tangents.

**(c) If |r _{1} - r_{2 }|< C_{1}C_{2 }< r_{1} + r_{2}**

There are 2 common tangents.

**(d) If C _{1}C_{2 }= |r_{1} - r_{2}|**

There is one common tangent.

**(e) If C _{1}C_{2 }<|r_{1} - r_{2}|**

There is no common tangent.

**16. Length of common tangent**

**(a) External tangent**

**(b) Internal tangent**

**17. The condition of orthogonality of two circles**

Two circles intersect each other at 90^{o} when

2g_{1}g_{2} + 2f_{1}f_{2} = c_{1}+c_{2}

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### Circle Notes for IIT-JEE, Download PDF!

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