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Construction of Incircle

C1 and C2 a...

Question

$C_{1}$ and $C_{2}$ are two non concentric fixed circle with radii $R_{1}$ and $R_{2}$ such that $R_{2}$ is contained in $C_{1}$ . A third circle $C$ moves in such a way that it touches $C_{1}$ internally and $C_{2}$ externally, then find the locus of its centre.

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Solution

$\begin{matrix} C C_{2} = r + r_{2} C C_{1} = r_{1} - r C C_{1} + C C_{2} = r_{1} + r_{2} (c o n s tan t) T h u s l o c u s o f C i s e l l i p s e . \end{matrix}$

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Q. Three circles

C_{1}, C_{2}

C_{3}

r_{1}, r_{2}

r_{1} + r_{2}

respectively are such that

C_{1}

C_{2}

touch each other externally and

C_{3}

internally. Another circle with radius

r_{3}

touches all the three circles. If

r_{1} > r_{2} > r_{3}

r_{1}, r_{2}, r_{3}

are in A.P. then

{(\frac{r_{1}}{r_{2}})}^{3}

A

: lf the circles

x^{2} + y^{2} = a^{2},

x^{2} + y^{2} - 6 x - 8 y + 9 = 0

touch externally then

a = 1

R

: Two circles with centres

C_{1}, C_{2}

r_{1}, r_{2}

respectively touch externally iff

C_{1} C_{2} = r_{1} + r_{2}

C_{1}

C_{2}

on Argand be given by

| z + 1 | = 3

| z - 2 | = 7

respectively. If a variable circle

| z - z_{0} | = r

be inside circle

C_{2}

such that it touches

C_{1}

C_{2}

internally then locus of '

Z_{0}

' describes a conic E whose eccentricity is equal to :

C_{1}

C_{2}

an Argand plane be given by

| z + 1 | = 3

| z - 2 | = 7

respectively. If a variable circle

| z - z_{0} | = r

be inside circle

C_{2}

such that it touches

C_{1}

C_{2}

internally then locus of

z_{0}

describes a conic

E

whose eccentricity is equal to

Q. Assertion :The circles

x^{2} + y^{2} + 2 a x + c = 0

x^{2} + y^{2} + 2 b y + c = 0

touch each other if

\frac{1}{c} = \frac{1}{a^{2}} + \frac{1}{b^{2}}

. Reason: Two cirlces with centre

c_{1}, c_{2}

r_{1}, r_{2}

touch each other if

c_{1} + c_{2} = r_{1} \pm r_{2}

.

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