CameraIcon

SearchIcon

MyQuestionIcon

1

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

Definite Integrals

ABC is a righ...

Question

$△ A B C$ is a right angled triangle. $O$ is the circumcenter of the $△ A B C$ and $R_{1}, R_{2}, R_{3}$ are the radii of the circumcircles of the triangle $O B A, O C A$ and $O A B$ respectively, then $\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}}$ is equal to

A

\frac{a b c}{R}

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

B

\frac{a b c}{R^{3}}

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

C

\frac{a b c}{R^{4}}

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

D

None of these

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

Open in App

Solution

The correct option is A $\frac{a b c}{R^{3}}$
$R_{1} = \frac{B C}{2 sin (∠ B O C)} = \frac{a}{2 sin 2 A} \Rightarrow \frac{a}{R_{1}} = 2 sin 2 A$

Similarly, $\frac{b}{R_{2}} = 2 sin 2 B$ and $\frac{c}{R_{3}} = 2 sin 2 C$

So, $\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} = 2 (sin 2 A + sin 2 B + sin 2 C)$

$= 2.4 sin A sin B sin C [∵ A + B + C = π]$

$= (2 sin A) (2 sin B) (2 sin C) = (\frac{a}{R}) (\frac{b}{R}) (\frac{c}{C}) = \frac{a b c}{R^{3}}$

Suggest Corrections

thumbs-up

0

Similar questions

Q. O is the circumcenter of

Δ A B C

R_{1}, R_{2}, R_{3}

are respectively the radii of the circumcircles of the triangles OBA,OCA and OAB

\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} = \frac{a b c}{R^{3}}

Q. O is the circumcentre of

△ A B C

R_{1}, R_{2}, R_{3}

are respectively, the radii of the circumcircles of the triangles OBC, OCA, and OAB. Prove that

\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} = \frac{a b c}{R^{3}}

O

is the circumcentre of the

△ A B C

R_{1}, R_{2}, R_{3}

are the radii of the circumcircles of the triangles

O B C, O C A

O A B

respectively, then

\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}}

Q. O is the circumcenter of

Δ ABC and R_{1}, R_{2} and R_{3}

are the radii of the circumcircles of the triangles OBC, OCA and OAB. Then

\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} =

O

is the circumcentre of

Δ A B C

R_{1}, R_{2}, R_{3}

are the radii of the circum-circle of triangle

O B C, O C A

O A B

respectively, then the value of

\frac{a}{R_{1}} + \frac{b}{R_{2}} + \frac{c}{R_{3}} = \frac{a b c}{R^{n}}

n

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Integration

PHYSICS

Watch in App

explore_more_icon

Explore more

Definite Integrals

Standard XII Physics

Join BYJU'S Learning Program

CrossIcon