JEE Main · 2006 · 30 Apr 2006 · Offline · Vector Algebra

The values of

a

, for which the points

A, B, C

with position vectors

2 \hat{i} - \hat{j} + \hat{k}, \hat{i} - 3 \hat{j} - 5 \hat{k}

and

a \hat{i} - 3 \hat{j} + \hat{k}

respectively are the vertices of a right-angled triangle with

C = \frac{π}{2}

are

Answer: (A)

Solution

The right angle is required at

C

, so

C A

and

C B

must be perpendicular. With

A = 2 \hat{i} - \hat{j} + \hat{k}

B = \hat{i} - 3 \hat{j} - 5 \hat{k}

and

C = a \hat{i} - 3 \hat{j} + \hat{k}

, we have

C A = A - C = (2 - a) \hat{i} + 2 \hat{j}

and

C B = B - C = (1 - a) \hat{i} - 6 \hat{k}

. Their dot product is

(2 - a) (1 - a) + 0 + 0

, and setting it to zero gives

(2 - a) (1 - a) = 0

. Hence the triangle is right-angled at

C

for

a = 2

and for

a = 1

JEE Main · 2006 · 30 Apr 2006 · Offline · Single correct option