Let be a real parameter and consider the points

MATHEMATICS · Multi-part · CBSE · Mathematics · Introduction to Three-Dimensional Geometry · Introduction to Three-Dimensional Geometry

Problem

Let

k

be a real parameter and consider the points

A(k,0,2)

B(0,k,2)

and

C(0,0,k)

in space. (i) Show that

AC=BC

for all real values of

k

, and hence identify the class of triangle

ABC

for arbitrary

k

.},{ (ii) Determine the value(s) of

k

(if any) for which

\triangle ABC

is equilateral. (iii) Investigate whether

\triangle ABC

can ever be degenerate for any real

k

, and justify your answer.