MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

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}}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

B

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

C

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

D

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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

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

Suggest Corrections

thumbs-up

0

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

S_{1} a n d S_{2}

touch internally to each other. Then the eccentricity of the locus of the centre of the curve

S

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}

Q. For the following reactions, the correct reaction rates

r_{1}, r_{2}, r_{1}^{'}

r_{2}^{'}

Let $c_{1} & c_{2}$ be the centers and $r_{1} & r_{2}$ be the radius of two circles. Then

Cases Conditions

p. 1. $| r_{1} - r_{2} | < c_{1} c_{2} < r_{1} + r_{2}$

q. 2. $| r_{1} - r_{2} | = c_{1} c_{2}$

r. 3. $c_{1} c_{2} < | r_{1} + r_{2} |$

s.

4. $r_{1} + r_{2} < c_{1} r_{2}$

t. 5. $r_{1} + r_{2} = c_{1} r_{2}$

Q. In a right angled triangle

B C = 5, A B = 4, A C = 3

S

be the circum circle. Let

S_{1}

be the circle touching both sides

A B

A C

S

internally. Let

S_{2}

be the circle touching the sides

A B

A C

△ A B C

and touching the circle

S

r_{1} r_{2}

are radii of circles

S_{1}

S_{2}

respectively then

r_{1} r_{2} =

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Hyperbola and Terminologies

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Definition of Hyperbola

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon