ABC is a triangle right-angled at B. If A(k,1,-1), B(2k,0,2) and C(2+2k,k,1) are the vertices of the triangle, then what is the value of k?

Explanation

The direction ratios of AB are (2k-k, 0-1, 2-(-1)) = (k, -1, 3). The direction ratios of BC are (2+2k-2k, k-0, 1-2) = (2, k, -1). Since \angle B = 90^{\circ}, the dot product of their direction ratios is zero: k(2) + (-1)(k) + 3(-1) = 0 \implies 2k - k - 3 = 0 \implies k = 3.

Related questions on 3D Geometry

• Consider the points A(2,4,6), B(-2,-4,-2), C(4,6,4) and D(8,14,12). Which of the following statements is/are correct? 1. The points ...
• Consider the equation of a sphere x^{2}+y^{2}+z^{2}-4x-6y-8z-16=0. Which of the following statements is/are correct? 1. z-axis is tangent ...
• A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
• Consider the following statements : 1. The direction ratios of y-axis can be \langle 0, 4, 0 \rangle 2. The direction ratios of a line <st...
• If L is the line with direction ratios \lt 3,-2, 6\gt and passing through (1,-1,1), then what are the coordinates of the points on $L...