JEE MAIN - Mathematics (2006 - No. 13)
The values of a, for which the points $$A, B, C$$ with position vectors $$2\widehat i - \widehat j + \widehat k,\,\,\widehat i - 3\widehat j - 5\widehat k$$ and $$a\widehat i - 3\widehat j + \widehat k$$ respectively are the vertices of a right angled triangle with $$C = {\pi \over 2}$$ are :
$$2$$ and $$1$$
$$-2$$ and $$-1$$
$$-2$$ and $$1$$
$$2$$ and $$-1$$
ব্যাখ্যা
$$\overrightarrow {CA} = \left( {2 - a} \right)\widehat i + 2\widehat j;$$
$$\overrightarrow {CB} = \left( {1 - a} \right)\widehat i - 6\widehat k$$
$$\overrightarrow {CA} .\overrightarrow {CB} = 0$$
$$\,\,\,\,\,\,\,\, \Rightarrow \left( {2 - a} \right)\left( {1 - a} \right) = 0$$
$$ \Rightarrow a = 2,1$$
$$\overrightarrow {CB} = \left( {1 - a} \right)\widehat i - 6\widehat k$$
$$\overrightarrow {CA} .\overrightarrow {CB} = 0$$
$$\,\,\,\,\,\,\,\, \Rightarrow \left( {2 - a} \right)\left( {1 - a} \right) = 0$$
$$ \Rightarrow a = 2,1$$