Question

Let A(x,y,z)A be a point in xy-plane, which is equidistant from three points (0,3,2),(2,0,3) and ( 0,0,1).

Let B=(1,4,−1) and C=(2,0,−2). Then among the statements the true statement(s) is/are:-

(S1) : △ABC is an isosceles right angled triangle, and

(S2) : the area of △ABC is \(9\sqrt 2 \over 2\)

Answer 1

Now \(A B=\sqrt{4+4+1}=3\)   

\(\mathrm{AC}=\sqrt{1+4+4}=3 \)

\( \mathrm{BC}=\sqrt{1+16+1}=\sqrt{18}\)     

\(\mathrm{AB}=\mathrm{AC}\)

isosceles \(\Delta\ \& \ \mathrm{AB}^{2}+\mathrm{AC}^{2}=\mathrm{BC}^{2}\)

right angle \(\Delta\)

Area of \(\triangle \mathrm{ABC}=\frac{1}{2} \times\) base.height

\(\frac{1}{2} \times 3 \times 3=\frac{9}{2}\)

So only \(\mathrm{S}_{1}\) is true