SATHEE: Vectors 1 Question 25

Vectors 1 Question 25

26. Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

(2001, 5M)

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Answer:

Correct Answer: 26. (b)

Solution:

Let $A D$ be the angular bisector of angle $A$ . Let $B C, A C$ and $A B$ are $α, β$ and $γ$ , respectively. Then, $\frac{B D}{D C} = \frac{γ}{β}$ .

Hence, position vector of $D = \frac{γ \vec{c} + β \vec{b}}{γ + β}$ . On $A D$ , there lies a point $I$ which divides it in ratio $γ + β : α$ .

Now, position vector of $I = \frac{α \vec{a} + β \vec{b} + γ \vec{c}}{α + β + γ}$

which is symmetric in $\vec{a}, \vec{b}, \vec{c}, α, β$ and $γ$ .

Hence $I$ lies on every angle bisector and angle bisectors are concurrent.

Here, $α = | \vec{b} - \vec{c} |, β = | \vec{a} - \vec{c} |, γ = | \vec{a} - \vec{b} |$ .

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Vectors 1 Question 25

26. Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

Answer:

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