Centres of two circles C₁ and C₂ each of unit radius are at a distance of 6 units from each othe

English

Gujarati

Hindi

10-1.Circle and System of Circles

normal

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$, then the radius of the circle $C$ is

$3$

$4$

$6$

$8$

(IIT-2009)

Solution

Let $A_1, A_2$ and $M$ be the centres of the circles $C_1, C_2$ and $C$ respectively. Let the common tangent through P to $C _1$ and C touch $C _1$ at $B _1$ and C at $B _2$ and it touches $C _2$ also at $B _2$

From right angled triangle $A _1 B_2, P$

$\text { if } \underline{ A _1 PB_2}=\alpha, \frac{ A _1 B_2}{A_1 P }=\frac{1}{3}$

$\Rightarrow \cos \alpha=\frac{2 \sqrt{2}}{3} \Rightarrow PB _1=2 \sqrt{2}= PB _2$

From triangle $MPB _2$

$\tan \alpha=\frac{ PB _2}{ MB _2}=\frac{2 \sqrt{2}}{ r }$

$\Rightarrow \frac{1}{2 \sqrt{2}}=\frac{2 \sqrt{2}}{ r }$

$\Rightarrow r =8$

Standard 11

Mathematics

Length of the tangent from $({x_1},{y_1})$ to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ is

normal

$x = 7$ touches the circle ${x^2} + {y^2} – 4x – 6y – 12 = 0$, then the coordinates of the point of contact are

easy

A line meets the co-ordinate axes in $A\, \& \,B. \,A$ circle is circumscribed about the triangle $OAB.$ If $d_1\, \& \,d_2$ are the distances of the tangent to the circle at the origin $O$ from the points $A$ and $B$ respectively, the diameter of the circle is :

normal

If the distances from the origin to the centres of the three circles ${x^2} + {y^2} – 2{\lambda _i}\,x = {c^2},(i = 1,\,2,\,3)$ are in $G. P.$, then the lengths of the tangents drawn to them from any point on the circle ${x^2} + {y^2} = {c^2}$ are in

hard

Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $AC$ is. . . . . .

easy

(IIT-2019)

Solution

Similar Questions

Length of the tangent from $({x_1},{y_1})$ to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ is

$x = 7$ touches the circle ${x^2} + {y^2} – 4x – 6y – 12 = 0$, then the coordinates of the point of contact are

A line meets the co-ordinate axes in $A\, \& \,B. \,A$ circle is circumscribed about the triangle $OAB.$ If $d_1\, \& \,d_2$ are the distances of the tangent to the circle at the origin $O$ from the points $A$ and $B$ respectively, the diameter of the circle is :

If the distances from the origin to the centres of the three circles ${x^2} + {y^2} – 2{\lambda _i}\,x = {c^2},(i = 1,\,2,\,3)$ are in $G. P.$, then the lengths of the tangents drawn to them from any point on the circle ${x^2} + {y^2} = {c^2}$ are in

Start a Free Trial Now