52  Circles

52.1 Introduction

A circle is the locus of all points in a plane that are equidistant from a fixed point called the center.
The fixed distance is called the radius.
Circles form the foundation for many problems in coordinate geometry and aptitude, especially in tangents, chords, geometry theorems, and mensuration.

52.2 Standard Equation of a Circle

For center \((h,k)\) and radius \(r\): \[ (x-h)^2+(y-k)^2=r^2 \]

Special cases: - Center at origin \((0,0)\): \(x^2+y^2=r^2\).
- If expanded: \(x^2+y^2+2gx+2fy+c=0\) represents a circle with center \((-g,-f)\) and radius \(r=\sqrt{g^2+f^2-c}\).

52.3 Key Properties

  • All radii are equal.
  • Diameter = \(2r\).
  • Circumference = \(2\pi r\).
  • Area = \(\pi r^2\).
  • Tangent to circle is perpendicular to radius at point of contact.
  • A line through center ⟂ to chord bisects chord.
  • Angle subtended by arc at center = 2 × angle subtended at circumference.

52.4 Equation of Tangent

At point \(P(x_1,y_1)\) on circle \(x^2+y^2=r^2\):
\[ xx_1+yy_1=r^2 \]

For circle \(x^2+y^2+2gx+2fy+c=0\):
Equation of tangent at \((x_1,y_1)\):
\[ xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0 \]

52.5 Equation of Normal

Normal at \(P(x_1,y_1)\): line through center \((h,k)\) and \(P(x_1,y_1)\).

52.6 Length of Tangent from External Point

From external point \(P(x_1,y_1)\) to circle \(x^2+y^2+2gx+2fy+c=0\):
\[ \text{Length}=\sqrt{x_1^2+y_1^2+2gx_1+2fy_1+c} \]

52.7 Power of a Point

For point \(P(x_1,y_1)\) and circle \(S\equiv x^2+y^2+2gx+2fy+c=0\):
\[ \text{Power}(P)=x_1^2+y_1^2+2gx_1+2fy_1+c \] If positive → point outside, zero → on circle, negative → inside.

52.8 Chord Properties

  • Equation of chord with midpoint \((x_1,y_1)\):
    Substitute \((x,y)\to(x+x_1,y+y_1)\) in circle equation, subtract circle eqn.
  • Perpendicular distance from center to chord = \(\sqrt{r^2-d^2}\), where \(d\) = half chord length.
  • Diameter subtends right angle on circle.

52.9 Circle and Line

Line \(ax+by+c=0\) intersects circle \(x^2+y^2=r^2\) if discriminant ≥ 0.
- Two real points: secant.
- One point: tangent.
- No intersection: line outside.

52.10 Circle and Circle

Two circles \(S_1\equiv x^2+y^2+2g_1x+2f_1y+c_1=0\), \(S_2\equiv x^2+y^2+2g_2x+2f_2y+c_2=0\):
- Radical axis: \(S_1-S_2=0\) (always a straight line).
- Two circles intersect if distance between centers ≤ sum of radii.
- Touch externally if distance = sum of radii.
- Touch internally if distance = |difference of radii|.

52.11 Circle Theorems (Geometry)

  1. Angle subtended by diameter = \(90^\circ\).
  2. Angles in same segment are equal.
  3. Opposite angles of cyclic quadrilateral sum to \(180^\circ\).
  4. Perpendicular from center to chord bisects it.

52.12 Solved Examples

Example 1
Find equation of circle with center \((2,-3)\) and radius 5.
\((x-2)^2+(y+3)^2=25\).

Example 2
Find equation of tangent to circle \(x^2+y^2=25\) at \((3,4)\).
\(3x+4y=25\).

Example 3
Find length of tangent from point \((7,1)\) to circle \(x^2+y^2=25\).
\(= \sqrt{7^2+1^2-25}=\sqrt{49+1-25}=\sqrt{25}=5\).

Example 4
Check whether line \(3x+4y=20\) touches circle \(x^2+y^2=25\).
Distance from center \((0,0)\) to line = \(|20|/\sqrt{3^2+4^2}=20/5=4\).
Since \(r=5\), line intersects at 2 points (secant), not tangent.

Example 5
Equation of circle passing through \((1,2),(3,-4),(5,6)\).
General form \(x^2+y^2+gx+fy+c=0\).
Substitute each point to form 3 equations, solve for \(g,f,c\).

52.13 Practice Problems

  1. Find equation of circle with diameter endpoints \((2,3)\) and \((4,7)\).
  2. Write equation of tangent to \(x^2+y^2=16\) at \((0,4)\).
  3. Find equation of circle passing through \((1,2),(2,3),(3,1)\).
  4. Find length of tangent from point \((5,12)\) to circle \(x^2+y^2=25\).
  5. Equation of common chord of \(x^2+y^2=25\) and \(x^2+y^2-6x+8=0\).
  6. Circle with center \((1,2)\) passes through \((4,6)\). Find equation.
  7. Two circles \(x^2+y^2=16\) and \((x-6)^2+y^2=9\) touch externally. Find point of contact.

52.14 Summary

  • Circle: locus of equidistant points from center.
  • General equation: \(x^2+y^2+2gx+2fy+c=0\).
  • Tangent ⟂ radius at point of contact.
  • Length of tangent, power of a point important for problem-solving.
  • Radical axis connects two circles.
  • Geometry theorems (angle in semicircle, cyclic quadrilateral) are exam favorites.