Question

O is the circumcentre of the triangle ABC and R₁, R₂, R₃ are the radii of the circumcircles of the triangle OBC, OCA and OAB respectively. Then $\frac { \mathrm { a } } { \mathrm { R } _ { 1 } }$+ $\frac { \mathrm { c } } { \mathrm { R } _ { 3 } }$is equal to –

• A. $\frac { \mathrm { abc } } { \mathrm { R } }$
• B.
• C.
• D.

Answer 1

Answer: (B)

Answer 2

We know that R = . Let D₁, D₂ and D₃ represent the

Areas of triangles OBC, OCA

and OAB respectively. Then

R₁=

̃ $\frac { 4 \Delta _ { 1 } } { \mathrm { R } ^ { 2 } }$

Similarly, $\frac { \mathrm { b } } { \mathrm { R } _ { 2 } }$ = $\frac { 4 \Delta _ { 2 } } { \mathrm { R } ^ { 2 } }$ and $\frac { \mathrm { c } } { \mathrm { R } _ { 3 } }$= $\frac { 4 \Delta _ { 3 } } { \mathrm { R } ^ { 2 } }$

\ $\frac { \mathrm { a } } { \mathrm { R } _ { 1 } }$+ $\frac { \mathrm { b } } { \mathrm { R } _ { 2 } }$ + $\frac { \mathrm { c } } { \mathrm { R } _ { 3 } }$ = (D₁+ D₂ + D₃)

= $\frac { 4 \Delta } { \mathrm { R } ^ { 2 } }$ = . $\frac { \mathrm { abc } } { 4 \mathrm { R } }$ = $\frac { a b c } { R ^ { 3 } }$