CameraIcon

SearchIcon

MyQuestionIcon

1

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

Parametric Equation of Parabola

Three circles...

Question

Three circles $C_{1}, C_{2}$ and $C_{3}$ with radii $r_{1}, r_{2}$ and $r_{1} + r_{2}$ respectively are such that $C_{1}$ and $C_{2}$ touch each other externally and $C_{3}$ internally. Another circle with radius $r_{3}$ touches all the three circles. If $r_{1} > r_{2} > r_{3}$ and $r_{1}, r_{2}, r_{3}$ are in A.P. then ${(\frac{r_{1}}{r_{2}})}^{3}$ is

Open in App

Solution

$cos (∠ C D A) = - cos (∠ C D B) \Rightarrow \frac{A D^{2} + D C^{2} - A C^{2}}{2 A D . D C} = - (\frac{B D^{2} + D C^{2} - B C^{2}}{2 B D . D C}) \Rightarrow \frac{(r_{1} + r_{2} - r_{1})^{2} + (r_{1} + r_{2} - r_{3})^{2} - (r_{1} + r_{3})^{2}}{2 r_{2} (r_{1} + r_{2} - r_{3})} = - \frac{(r_{1})^{2} + (r_{1} + r_{2} - r_{3})^{2} - (r_{2} + r_{3})^{2}}{2 r_{1} (r_{1} + r_{2} - r_{3})} 2 r_{2} = r_{1} + r_{3} as they are in A.P \Rightarrow \frac{(r_{2})^{2} + (2 r_{1} - r_{2})^{2} - (2 r_{2})^{2}}{2 r_{2} (2 r_{1} - r_{2})} = - \frac{(r_{1})^{2} + (2 r_{1} - r_{2})^{2} - (3 r_{2} - r_{1})^{2}}{2 r_{1} (2 r_{1} - r_{2})}$
By solving this we get
${(\frac{r_{1}}{r_{2}})}^{3} = 2$

Suggest Corrections

thumbs-up

0

Similar questions

Q. Three Circles with centres

C_{1}, C_{2}, C_{3}

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

r_{1} < r_{2} < r_{3}

are placed as shown in the figure.
Then,

r_{2} =

Q. Five circles

C_{1}, C_{2}, C_{3}, C_{4}, C_{5}

r_{1}, r_{2}, r_{3}, r_{4}, r_{5}

(r_{1} < r_{2} < r_{3} < r_{4} < r_{5})

C_{i}

C_{i} + 1

touch each other externally for all

i = 1, 2, 3, 4

. If all the five circles touches two straight lines

L_{1}

L_{2}

r_{1} = 2

r_{5} = 32

r_{3}

Q. Three circles of radii

r_{1}

r_{2}

r_{3}

are drawn concentric to each other. The radii

r_{1}

r_{2}

are such that the area of the circle with radius

r_{1}

is equal to the area between the circles of radii

r_{2}

r_{1}

. The area between the circles of radii

r_{3}

r_{2}

is equal to the area between the circles of radii

r_{2}

r_{1}

What is the value of

r_{1} : r_{2} : r_{3}

?

Q. There are three circles

C_{1}, C_{2}, C_{3}

r_{1}, r_{2}, r_{3} (r_{1} < r_{2} < r_{3})

respectively touching each other externally as shown in the figure and

L_{1}, L_{2}

are two common tangents to these circles from point

P

Which of the following is(are) correct?

Q. Three circles of radii

r_{1}, r_{2}

r_{3}

are drawn concentric to each other. The radii

r_{1}

r_{2}

are such that the area of the circle with radius

r_{1}

is equal to the area between the circles of radius

r_{2}

r_{1}

. The area between the circles of radii

r_{3}

r_{2}

is equal to area between the circles of radii

r_{2}

r_{r}

. What is the value of

r_{1} : r_{2} : r_{3} ?

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Parametric Representation

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Parametric Equation of Parabola

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon