Use these Inter 2nd Year Maths 2B Formulas PDF Chapter 1 Circle to solve questions creatively.

## Intermediate 2nd Year Maths 2B Circle Formulas

→ The locus of a point in a plane such that its distance from a fixed point in the plane is always the same is called a circle.

→ The equation of a circle with centre (h, k) and radius r is (x – h)^{2} + (y – k)^{2} = r^{2}

→ The equation of a circle in standard form is x^{2} + y^{2} = r^{2}.

→ The equation of a circle in general form is x^{2} + y^{2} + 2gx + 2fy + c = 0 and its centre is (-g, -f), radius is \(\sqrt{g^{2}+f^{2}-c}\).

→ The intercept made by x^{2} + y^{2} + 2gx + 2fy + c = 0

- on X-axis is 2\(\sqrt{g^{2}-c}\) if g
^{2}> c. - on Y-axis is 2\(\sqrt{f^{2}-c}\) if f
^{2}> c.

→ If the extremities of a diameter of a circle are (x_{1}, y_{1}) and (x_{2}, y_{2}) then its equation is (x – x_{1}) (x – x_{2}) + (y – y_{1}) (y – y_{2}) = 0

→ The equation of a circle passing through three non-collinear points (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is

\(\left|\begin{array}{lll}

x_{1} & y_{1} & 1 \\

x_{2} & y_{2} & 1 \\

x_{3} & y_{3} & 1

\end{array}\right|\) = (x^{2} + y^{2}) + \(\left|\begin{array}{lll}

c_{1} & y_{1} & 1 \\

c_{2} & y_{2} & 1 \\

c_{3} & y_{3} & 1

\end{array}\right|\) x + \(\left|\begin{array}{lll}

x_{1} & C_{1} & 1 \\

x_{2} & C_{2} & 1 \\

x_{3} & C_{3} & 1

\end{array}\right|\) y + \(\left|\begin{array}{lll}

x_{1} & y_{1} & C_{1} \\

x_{2} & y_{2} & C_{2} \\

x_{3} & y_{3} & C_{3}

\end{array}\right|\) = 0.

where c_{i} = – (x_{i}^{2} + y_{i}^{2})

→ The centre of the circle passing through three non-collinear points (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is

\(\left[\frac{\left|\begin{array}{lll}

c_{1} & y_{1} & 1 \\

c_{2} & y_{2} & 1 \\

c_{3} & y_{3} & 1

\end{array}\right|}{(-2)\left|\begin{array}{lll}

x_{1} & y_{1} & 1 \\

x_{2} & y_{2} & 1 \\

x_{3} & y_{3} & 1

\end{array}\right|}, \frac{\left|\begin{array}{lll}

x_{1} & c_{1} & 1 \\

x_{2} & c_{2} & 1 \\

x_{3} & c_{3} & 1

\end{array}\right|}{(-2)\left|\begin{array}{lll}

x_{1} & y_{1} & 1 \\

x_{2} & y_{2} & 1 \\

x_{3} & y_{3} & 1

\end{array}\right|}\right]\)

→ The parametric equations of a circle with centre (h, k) and radius (r ≥ 0) are given by

x = h + r cos θ

y = k + r sin θ 0 ≤ 6 < 2π.

→ A point P(x_{1}, y_{1}) is an interior point or on the circumference or an exterior point of a circles S = 0 ⇔ S_{11} \(\frac{<}{>}\) 0.

→ The power of P(x_{1}, y_{1}) with respect to the circle S = 0 is S_{11}.

→ A point P(x_{1}, y_{1}) is an interior point or on the circumference or exterior point of the circle S = 0 ⇔ the power of P with respect to S = 0 is negative, zero and positive.

→ If a straight line through a point P(x_{1}, y_{1}) meets the circle S = 0 at A and B then the power of P is equal to PA. PB.

→ The length of the tangent from P(x_{1}, y_{1}) to S = 0 is \(\sqrt{S_{11}}\).

→ The straight line l = 0 intersects, touches or does not meet the circles = 0 according as l < r, l = r or l > r where l is the perpendicular distance from the centre of the circle to the line l = 0 and r is the radius.

→ For every real value of m the straight line y = mx ± r \(\sqrt{1+m^{2}}\) is a tangent to the circle x^{2} + y^{2} = r^{2}.

→ If r is the radius of the circle S = x^{2} + y^{2} + 2gx + 2fy + c = 0 then for every real value of m the straight line y + f = m(n + g) ±r + m2 will be a tangent to the circle.

→ If P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) are two points on the circle S = 0 then the secant’s \((\stackrel{\leftrightarrow}{P Q})\) equation is S_{1} + S_{2} = S_{12}

→ The equation of tangent at (x_{1}, y_{1}) of the circle S = 0 is S_{1} = 0.

→ If θ_{1}, θ_{2} are two points on S = x^{2} + y^{2} + 2gx + 2fy + c = 0 then the equation of the chord joining the points θ_{1}, θ_{2} is

(x + g) cos \(\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\) + (y + f) sin \(\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\) = r cos \(\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\)

→ The equation of the tangent at θ of the circle S = 0 is (x + g) cos θ + (y + f) sin θ = r.

→ The equation of normal at (x_{1}, y_{1}) of the circle

S = 0 is (x – x_{1}) (y_{1} + f) – (y – y_{1}) (x_{1} + g) = 0.

→ The chord of contact of P(x_{1} y_{1}) (exterior point) with respect to S = 0 is S_{1} = 0.

→ The equation of the polar of a point P(x_{1}, y_{1}) with respect to S = 0 is S_{1} = 0.

P(x_{1}, y_{1}) |
Tangent at P | Chord of contact at p | Polar of P |

(i) Interior of the circle | Does not exist | Does not exist (not defined) |
S_{1} = 0(P is different from the centre of the circle) |

(ii) On the circle | S_{1} = 0 |
S_{1} = 0 |
S_{1} = 0 |

(iii) Exterior of the circle | Does not exist | S_{1} = 0 |
S_{1} = 0 |

→ The pole of lx + my + n = 0 with respect to S = 0 is

\(\left(-g+\frac{l r^{2}}{l g+m f-n},-f+\frac{m r^{2}}{l g+m f-n}\right)\)

→ Where r is the radius of the circle. The polar of P(x_{1}, y_{1}) with respect to S = 0 passes through Q(x_{2}, y_{2}) ⇔ the polar of Q with respect to S – 0 passes through P.

→ The points (x_{1}, y_{1}) and (x_{2}, y_{2}) are conjugate points with respect to S = 0 if S_{12} = 0

→ Two lines l_{1}x = m_{1}y + n_{1} = 0, l_{2}x + m_{2}y + n_{2} = 0 are conjugate with respect to x^{2} + y^{2} = a^{2} ⇔ (l_{1}l_{2} + m_{1}m_{2}) = n_{1}n_{2}

→ Two points P, Q are said to be inverse points with respect to S = 0 if CP. CQ = r^{2} where C is the centre and r is the radius of the circle S = 0.

→ If (x_{1}, y_{1}) is the mid-point of a chord of the circle S = 0 then its chord equation is S_{1} = S_{11}.

→ The pair of common tangents to the circles S = 0, S’ = 0 touching at a point on the lines segment \(\overline{\mathrm{C}_{1} \mathrm{C}_{2}}\) (C_{1}, C_{2} are centres of the circles) is called transverse pair of common tangents.

→ The pair of common tangents to the circles S = 0, S’ = 0 intersecting at a point not in \(\overline{\mathrm{C}_{1} \mathrm{C}_{2}}\) is called as direct pair of common tangents.

→ The point of intersection of transvese (direct) common tangents is called internal (external) Centre of similitude.

Situation | No of common tangents |

1. \( \overline{C_{1} C_{2}} \) > r_{1} + r_{2} |
4 |

2. r_{1} + r_{2} = \( \overline{C_{1} C_{2}} \) |
3 |

3. |r_{1} – r_{2}| < \( \overline{C_{1} C_{2}} \) < r_{1} + r_{2} |
2 |

4. C_{1}C_{2} = |r_{1} – r_{2}| |
1 |

5. C_{1}C_{2} < |r_{1} – r_{2}| |
0 |

→ The combined equation of the pair of tangents drawn from an external point P(x_{1}, y_{1}) to the circle S = 0 is SS_{11} = S^{2}_{1}.

Equation of a Circle:

The equation of the circle with centre C (h, k) and radius r is (x – h)^{2} + (y – k)^{2} = r^{2}.

Proof:

Let P(x_{1}, y_{1}) be a point on the circle.

P lies in the circle ⇔ PC = r ⇔ \(\sqrt{\left(\mathrm{x}_{1}-\mathrm{h}\right)^{2}+\left(\mathrm{y}_{1}-\mathrm{k}\right)^{2}}\) = r

⇔ (x_{1} – h)^{2} + (y_{1} – k)^{2} = r^{2}.

The locus of P is (x – h)^{2} + (y – k)^{2} = r^{2}.

∴ The equation of the circle is (x – h)^{2} + (y – k)^{2} = r^{2}.

Note: The equation of a circle with centre origin and radius r is (x – 0)^{2} + (y – 0)^{2} = r^{2}

i.e., x^{2} + y^{2} = r^{2} which is the standard equation of the circle.

Note: On expanding equation (1), the equation of a circle is of the form x^{2} + y^{2} + 2gx + 2fy + c = 0.

Theorem: If g^{2} + f^{2} – c ≥ 0, then the equation x^{2} + y^{2} + 2gx + 2fy + c = 0 represents a circle with centre (- g, – f) and radius \(\sqrt{g^{2}+f^{2}-c}\).

Note: If ax^{2} + ay^{2} + 2gx + 2fy + c = 0 represents a circle, then its centre = \(\left(-\frac{g}{a},-\frac{f}{a}\right)\) and its radius \(\frac{\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{ac}}}{|\mathrm{a}|}\).

Theorem: The equation of a circle having the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) as diameter is (x – x_{1}) (x – x_{2}) + (y – y_{1}) (y – y_{2}) = 0.

Let P(x, y) be any point on the circle. Given points A(x_{1}, y_{1}) and B(x_{2}, y_{2}).

Now ∠APB = \(\frac{\pi}{2}\). (Angle in a semi circle.)

Slope of AP. Slope of BP = – 1

⇒ \(\frac{y-y_{1}}{x-x_{1}} \frac{y-y_{2}}{x-x_{2}}\) = – 1

⇒ (y – y_{2}) (y – y_{1}) = – (x – x_{2}) (x – x_{1}) = 0

⇒ (x – x_{2}) (x – x_{1}) + (y – y_{2}) (y – y_{1}) = 0

Definition: Two circles are said to be concentric if they have same center.

The equation of the circle concentric with the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 is of the form x^{2} + y^{2} + 2gx + 2fy + k = 0.

The equation of the concentric circles differs by constant only.

Parametric Equations of A Circle:

Theorem: If P(x, y) is a point on the circle with centre C(α, β) and radius r, then x = α + r cosθ, y = β + r sin θ where 0 ≤ θ < 2π.

Note: The equations x = α + r cos θ, y = + r sin θ, 0 ≤ θ < 2π are called parametric equations of the circle with centre (α, β) and radius r.

Note: A point on the circle x^{2} + y^{2} = r^{2} is taken in the form (r cos θ, r sin θ). The point (r cos θ, r sin θ) is simply denoted as point θ.

Theorem:

(1) If g^{2} – c > 0 then the intercept made on the x axis by the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 is 2\(\sqrt{g^{2}-a c}\)

(2) If f^{2} – c >0 then the intercept made on the y axis by the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 is 2\(\sqrt{f^{2}-b c}\)

Note: The condition for the x-axis to touch the circle

x^{2} + y^{2} + 2gx + 2fy + c = 0 (c > 0) is g^{2} = c.

Note: The condition of the y-axis to touch the circle

x^{2} + y^{2} + 2gx + 2fy + c = 0 (c > 0) is f^{2} = c.

Position of Point:

Let S = 0 be a circle and P(x_{1}, y_{1}) be a point I in the plane of the circle. Then

- P lies inside the circle S = 0 ⇔ S
_{11}< 0 - P lies in the circle S = 0 ⇔ S
_{11}= 0 - Plies outside the circle S = 0 ⇔ S
_{11}= 0

Power of a Point:

Let S = 0 be a circle with centre C and radius r. Let P be a point. Then CP^{2} – r^{2} is called power of P with respect to the circle S = 0.

Theorem: The power of a point P(x_{1}, y_{1}) with respect to the circle S = 0 is S_{11}.

Theorem: The length of the tangent drawn from an external point P(x_{1}, y_{1}) to the circle s = 0 is \(\sqrt{\mathrm{S}_{11}}\).

Theorem: The equation of the tangent to the circle S = 0 at P(x_{1}, y_{1}) is S_{1} = 0.

Theorem: The equation of the normal to the circle S = x^{2} + y^{2} + 2gx + 2fy + c = 0 at P(x_{1}, y_{1}) is

(y_{1} + f) (x – x_{1}) – (x_{1} + g) (y – y_{1}) = 0.

Corollary: The equation of the normal to the circle x^{2} + y^{2} = a^{2} at P(x_{1}, y_{1}) is y_{1}x – x_{1}y = 0.

Theorem: The condition that the straight line lx + my + n = 0 may touch the circle x^{2} + y^{2} = a^{2} is n^{2} = a^{2}(l^{2} + m^{2}) and the point of contact is \(\left(\frac{-a^{2} 1}{n}, \frac{-a^{2} m}{n}\right)\).

Proof:

The given line is lx + my + n = 0 …… (1)

The given circle is x^{2} + y^{2} = r^{2} ……. (2)

Centre C = (0, 0), radius = r

Line (1) is a tangent to the circle (2)

⇔ The perpendicular distance from the centre C to the line (1) is equal to the radius r.

⇔ \(\left|\frac{0-n}{\sqrt{1^{2}+m^{2}}}\right|\) = r

⇔ (n)^{2} = r^{2} (l^{2} + m^{2})

Let P(x_{1}, y_{1}) be the point of contact.

Equation of the tangent is S_{1} = 0, ⇒ x_{1}x + y_{1}y – r^{2} = 0. —- (3)

Equations (1) and (3) are representing the same line, therefore, \(\frac{x_{1}}{l}=\frac{y_{1}}{m}=\frac{-a^{2}}{n}\) ⇒ x_{1} = \(\frac{-a^{2} l}{n}\), y_{1} = \(\frac{-a^{2} m}{n}\)

Therefore, point of contact is \(\left(\frac{-a^{2} l}{n}, \frac{-a^{2} m}{n}\right)\)

Theorem: The condition for the straight line lx + my + n = 0 may be a tangent to the circle

x^{2} + y^{2} + 2gx + 2fy + c = 0 is (g^{2} + f^{2} – c) (l^{2} + m^{2}) = (lg + mf – n)^{2}.

Proof:

The given line is lx + my + n = 0 …….. (1)

The given circle is x^{2} + y^{2} + 2gx + 2fy + c = 0 …….. (2)

Centre C = (- g, – f), radius r = \(\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{c}}\)

Line (1) is a tangent to the circle (2)

⇔ The perpendicular distance from the centre C to the line (1) is equal to the radius r.

⇔ \(\left|\frac{-\lg -m f+c}{\sqrt{1^{2}+m^{2}}}\right|\) = \(\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{c}}\)

⇔ (lg + mf -n)^{2} = (g^{2} + f^{2} – c) (l^{2} + m^{2})

Corollary: The condition for the straight line y = mx + c to touch the circle

x^{2} + y^{2} = r^{2} is c^{2} = r^{2}(1 + m^{2}).

The given line is y = mx + c i.e., mx – y + c = 0 … (1)

The given circle is S = x^{2} + y^{2} = r^{2}

Centre C = (0,0), radius = r.

If (1) is a tangent to the circle, then

Radius of the circle = perpendicular distance from centre of the circle to the line.

⇒ r = \(\frac{|c|}{\sqrt{m^{2}+1}}\) ⇒ r^{2} = \(\frac{c^{2}}{m^{2}+1}\) ⇒ r^{2} (m^{2} + 1) = c^{2}

Corollary: If the straight line y = mx + c touches the circle x^{2} + y^{2} = r^{2}, then their point of contact is \(\left(-\frac{r^{2} m}{c}, \frac{r^{2}}{c}\right)\).

Proof:

The given line is y = mx + c i.e., mx – y + c = 0 ……. (1)

The given circle is S = x^{2} + y^{2} = r^{2} ……. (2)

Centre C = (0, 0), radius = r

Let P(x_{1}, y_{1}) be the point of contact.

Equation of the tangent is S_{1} = 0, ⇒ x_{1}x + y_{1}y – r^{2} = 0. ………. (3)

Equations (1) and (3) are representing the same line, therefore, \(\frac{x_{1}}{m}=\frac{y_{1}}{-1}=\frac{-r^{2}}{c}\) ⇒ x_{1} = \(\frac{-r^{2} m}{c}\), y_{1} = \(\frac{r^{2}}{c}\)

Point of contact is (x_{1}, y_{1}) = \(\left(-\frac{\mathrm{r}^{2} \mathrm{~m}}{\mathrm{c}}, \frac{\mathrm{r}^{2}}{\mathrm{c}}\right)\)

Theorem: If P(x, y) is a point on the circle with centre C(α, β) and radius r, then x = α + r cos θ, y = β + r sin θ where 0 ≤ θ < 2π.

Note 1: The equations x = α + r cos θ, y = β + r sin θ, 0 ≤ θ < 2π are called parametric equations of the circle with centre (α, β) and radius r.

Note 2: A point on the circle x^{2} + y^{2} = r^{2} is taken in the form (r cosθ, r sin θ). The point (r cosθ, r sin θ) is simply denoted as point θ.

Theorem: The equation of the chord joining two points θ_{1} and θ_{2} on the circle

x^{2} + y^{2} + 2gx + 2fy + c = 0 is (x + g)cos\(\frac{\theta_{1}+\theta_{2}}{2}\) + (y + f) sin \(\frac{\theta_{1}+\theta_{2}}{2}\) = r cos \(\frac{\theta_{1}+\theta_{2}}{2}\) where r is the radius of the circle.

Note 1: The equation of the chord joining the points θ_{1} and θ_{2} on the circle x^{2} + y^{2} = r^{2} is x cos\(\frac{\theta_{1}+\theta_{2}}{2}\) + y sin\(\frac{\theta_{1}+\theta_{2}}{2}\) = r cos\(\frac{\theta_{1}-\theta_{2}}{2}\)

Note 2: The equation of the tangent at P(θ) on the circle (x + g) cos θ + (y + f) sin θ = \(\sqrt{g^{2}+f^{2}-c}\).

Note 3: The equation of the tangent at P(θ) on the circle x^{2} + y^{2} = r^{2} is x cos θ + y sin θ = r.

Note 4: The equation of the normal at P(θ) on the circle x^{2} + y^{2} = r^{2} is x sin θ – y cos θ = r.

Theorem:

If a line passing through a point P(x_{1}, y_{1}) intersects the circle S = 0 at the points A and B then PA.PB = |S_{11}|.

Corollary:

If the two lines a_{1}x + b_{1}y + c_{1} = 0, a_{2}x + b_{2}y + c_{2} = 0 meet the coordinate axes in four distinct points then those points are concyclic ⇔ a_{1}a_{2} = b_{1}b_{2}.

Corollary:

If the lines a_{1}x + b_{1}y + c_{1} = 0, a_{2}x + b_{2}y + c_{2} = 0 meet the coordinate axes in four distinct concyclic points then the equation of the circle passing through these concyclic points is (a_{1}x + b_{1}y + c_{1}) (a_{2}x + b_{2}y + c_{2}) – (a_{1}b_{2} + a_{2}b_{1})xy = 0.

Theorem:

Two tangents can be drawn to a circle from an external point.

Note:

If m_{1}, m_{1} are the slopes of tangents drawn to the circle x^{2} + y^{2} = a^{2} from an external point (x_{1}, y_{1}) then m_{1} + m_{2} = \(\frac{2 x_{1} y_{1}}{x_{1}^{2}-a^{2}}\), m_{1}m_{2} = \(\frac{y_{1}^{2}-a^{2}}{x_{1}^{2}-a^{2}}\).

Theorem:

If θ is the angle between the tangents through a point P to the circle S = 0 then tan = \(\frac{\theta}{2}=\frac{r}{\sqrt{S_{11}}}\) where r is the radius of the circle.

Proof:

Let the two tangents from P to the circle S = 0 touch the circle at Q, R and θ be the angle between

these two tangents. Let C be the centre of the circle. Now QC = r, PQ = \(\sqrt{S_{11}}\) and ∆PQC is a right angled triangle at Q.

∴ tan \(\frac{\theta}{2}=\frac{\mathrm{QC}}{\mathrm{PQ}}=\frac{\mathrm{r}}{\sqrt{\mathrm{S}_{11}}}\)

Theorem: The equation to the chord of contact of P(x_{1}, y_{1}) with respect to the circle S = 0 is S_{1} = 0.

Theorem: The equation of the polar of the point P(x_{1}, y_{1}) with respect to the circle S = 0 is S_{1} = 0.

Theorem: The pole of the line lx + my + n = 0 (n ≠ 0) with respect to x^{2} + y^{2} = a^{2} is \(\left(-\frac{1 a^{2}}{n},-\frac{m a^{2}}{n}\right)\)

Proof :

Let P(x_{1}, y_{1}) be the pole of lx + my + n = 0 ……. (1)

The polar of P with respect to the circle is:

xx_{1} + yy_{1} – a^{2} = 0

Now (1) and (2) represent the same line

∴ \(\frac{\mathrm{x}_{1}}{\ell}=\frac{\mathrm{y}_{1}}{\mathrm{~m}}=\frac{-\mathrm{a}^{2}}{\mathrm{n}}\) ⇒ x_{1} = \(\frac{-\mathrm{la}^{2}}{\mathrm{n}}\), y_{1} = \(\frac{-\mathrm{ma}^{2}}{\mathrm{n}}\)

∴ Pole P = \(\left(-\frac{1 a^{2}}{n},-\frac{m a^{2}}{n}\right)\)

Theorem: If the pole of the line lx + my + n = 0 with respect to the circle x^{2} + y^{2} + 2gx + 21y + c = 0 is (x_{1}, y_{1}) then \(\frac{x_{1}+g}{\ell}=\frac{y_{1}+f}{m}=\frac{r^{2}}{\lg +\mathrm{mf}-\mathrm{n}}\) where r is the radius of the circle.

Proof:

Let P(x_{1}, y_{1}) be the pole of the line lx + my + n = 0 ……. (1)

The poiar of P with respect to S = 0 is S_{1} = 0

xx_{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0

⇒ (x_{1} + g)x + (y_{1} + f) + gx_{1} + fy_{1} + c = 0 …….. (2)

Now (1) and (2) represent the same line.

∴ \(\frac{\mathrm{x}_{1}+\mathrm{g}}{\ell}=\frac{\mathrm{y}_{1}+\mathrm{f}}{\mathrm{m}}=\frac{\mathrm{gx} \mathrm{x}_{1}+\mathrm{gy} \mathrm{y}_{1}+\mathrm{c}}{\mathrm{n}}\) = k(say)

\(\frac{\mathrm{x}_{1}+\mathrm{g}}{\ell}\) = k ⇒ x_{1} + g = l k ⇒ x_{1} = lk – g

\(\frac{\mathrm{y}_{1}+\mathrm{f}}{\mathrm{m}}\) = k ⇒ y_{1} + f = m k ⇒ y_{1} = mk – f

\(\frac{g x_{1}+g y_{1}+c}{n}\) = k ⇒ gx_{1} + gy_{1} + c = nk

⇒ g(lk – g) + f(mk – f) + c = nk

⇒ k (lg + mf – n) = g^{2} + f^{2} – c = r^{2} Where r is the radius of the circle ⇒ k = \(\frac{r^{2}}{\lg +\mathrm{mf}-\mathrm{n}}\)

∴ \(\frac{\mathrm{x}_{1}+\mathrm{g}}{\ell}=\frac{\mathrm{y}_{1}+\mathrm{f}}{\mathrm{m}}=\frac{\mathrm{r}^{2}}{\lg +\mathrm{mf}-\mathrm{n}}\)

Theorem: The lines l_{1}x + m_{1}y + n_{1} = 0 and l_{2}x + m_{2}y + n_{1}y = 0 are conjugate with respect to the circle x^{2} + y_{1} + 2gx + 2fy + c = 0 iffr_{1} (l_{1}l_{2} + m_{1}m_{2}) = (l_{1}g + m_{1}f – n_{1}) (l_{2}g + m_{2}f – n_{2}).

Theorem: The condition for the lines l_{1}x + m_{1}y + n_{1} = 0 and l_{2}x + m_{2}y + n_{2} = 0 to be conjugate with respect to the circle x^{2} + y^{2} = a^{2} is a^{2}(l_{1}l_{2} + m_{1}m_{2}) = n_{1}n_{2}.

Theorem: The equation of the chord of the circle S = 0 having P(x_{1}, y_{1}) as its midpoint is S_{1} = S_{11}.

Theorem: The length of the chord of the circle S = 0 having P(x_{1}, y_{1}) as its midpoint is 2\(\sqrt{\left|S_{11}\right|}\).

Theorem: The equation to the pair of tangents to the circle

S = 0 from P(x_{1}, y_{1}) is S^{2}_{1} = S_{11}S.

Proof:

Let the tangents from P to the circle S = 0 touch the circle at A and B.

Equation of AB is S_{1} = 0.

i.e., x_{1}x + y_{1}y + g(x + x_{1}) + f(y + y_{1}) + c = 0 ———- (i)

Let Q(x_{2}, y_{2}) be any point on these tangents. Now locus of Q will be the equation of the pair of tangents drawn from P.

The line segment PQ is divided by the line AB in the ratio – S_{11}:S_{22}

⇒ \(\frac{P B}{Q B}=\left|\frac{S_{11}}{S_{22}}\right|\) ———— (ii)

But PB = \(\sqrt{S_{11}}\), QB = \(\sqrt{S_{22}}\) ⇒ \(\frac{P B}{Q B}=\frac{\sqrt{S_{11}}}{\sqrt{S_{22}}}\) ———— (iii)

From (ii) and (iii) ⇒ \(\frac{s_{11}^{2}}{s_{22}^{2}}=\frac{S_{11}}{S_{22}}\)

⇒ S_{11}S_{22} = S^{2}_{12}

Hence locus of Q(x_{2}, y_{2}) is S_{11}S = S^{2}_{12}

Touching Circles: Two circles S = 0 and S’ = 0 are said to touch each other if they have a unique point P in common. The common point P is called point of contact of the circles S = 0 and S’ = 0.

Circle – Circle Properties: Let S = 0, S’ = 0 be two circle with centres C_{1}, C_{2} and radii r_{1}, r_{2} respectively.

- If C
_{1}C_{2}> r_{1}+ r_{2}then each circle lies completely outside the other circle. - If C
_{1}C_{2}= r_{1}+ r_{2}then the two circles touch each other externally. The point of contact divides C_{1}C_{2}in the ratio r_{1}: r_{2}internally. - If |r
_{1}– r_{2}| < C_{1}C_{2}< r_{1}+ r_{2}then the two circles intersect at two points P and Q. The chord \(\overline{\mathrm{PQ}}\) is called common chord of the circles. - If C
_{1}C_{2}= |r_{1}– r_{2}| then the two circles touch each other internally. The point of contact divides C_{1}C_{2}in the ratio r_{1}: r_{2}externally. - If C
_{1}C_{1}< |r_{1}– r_{2}] then one circle lies completely inside the other circle.

Common Tangents: A line L = 0 is said to be a common tangent to the circle S = 0, S’ = 0 if L = 0 touches both the circles.

Definition: A common tangent L = 0 of the circles S = 0, S’= 0 is said to be a direct common tangent of the circles if the two circles S = 0, S’ = 0 lie on the same side of L = 0.

Centres of Similitude:

Let S = 0, S’ = 0 be two circles.

- The point of intersection of direct common tangents of S = 0, S’ = 0 is called external centre of similitude.
- The point of intersection of transverse common tangents of S = 0, S’ = 0 is called internal centre of similitude.

Theorem:

Let S = 0, S’ = 0 be two circles with centres C_{1}, C_{2} and radii r_{1}, r_{2} respectively. If A_{1} and A_{2} are respectively the internal and external centres of similitude circles s = 0, S’ = 0 then

- A
_{1}divides C_{1}C_{2}in the ratio r_{1}: r_{2}internally. - A
_{2}divides C_{1}C_{2}in the ratio r_{1}: r_{2}internally.