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Definition of Hyperbola

A variable ci...

Question

A variable circles $S$ touches $S_{1} : | z - z_{1} | = r_{1}$ internally and $S_{2} : | z - z_{2} | = r_{2}$ externally while the curves $S_{1}$ and $S_{2}$ touch internally to each other. Then the eccentricity of the locus of the centre of the curve S is equal to

A

\frac{r_{1}}{r_{2}}

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B

\frac{r_{2}}{r_{1}}

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C

\frac{r_{1} + r_{2}}{r_{1} - r_{2}}

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D

\frac{r_{1} - r_{2}}{r_{1} + r_{2}}

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Solution

The correct option is D $\frac{r_{1} - r_{2}}{r_{1} + r_{2}}$
When two circle touches externally as shown in fig (i)
When two circle touches internally as shown in fig (ii)
A variable circle S touches $S_{1} : | z - z_{2} | = r_{1}$ internally and $S_{1} : | z - z_{2} | = r_{1}$ externally
A necessary and sufficient condition for the two circles to intersect at two distinct points is
where $C_{1}$ ,{C}_{2} $a r e t h e c e n t e r s a n d r_{1} r_{2}$ be the radii of two circles.
If two circles with centers $C_{1} (x_{1}, y_{1}) a n d C_{2} (x_{2}, y_{2})$ and radii $r_{1} a n d r_{2}$ respectively , touch each other externally,
$= [(\frac{r_{1} x_{2} + r_{2} x_{1}}{r_{1} + r_{2}})], [(\frac{r_{1} y_{2} + r_{2} y_{1}}{r_{1} + r_{2}})]$
The circle touch each other internally if $C_{1} C_{2} = r_{1} - r_{2}$
The circle touch each other externally if $C_{1} C_{2} = r_{1} + r_{2}$
$∴$ Eccentricity of the locus of the center = $\frac{r_{1} - r_{2}}{r_{1} + r_{2}}$

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Similar questions

Q. If a variable circle

S

S_{1} : | z - z_{1} | = r_{1}

S_{2} : | z - z_{2} | = r_{2}

externally while the curves

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