Q53 / 150 · MHT CET 2008 · Mathematics
The line joining the points $6 \overrightarrow{\mathbf{a}}-4 \overrightarrow{\mathbf{b}}+4 \overrightarrow{\mathbf{c}},-4 \overrightarrow{\mathbf{c}}$ and the line joining the
points $-\overrightarrow{\mathbf{a}}-2 \overrightarrow{\mathbf{b}}-3 \overrightarrow{\mathbf{c}}, \overrightarrow{\mathbf{a}}+2 \overrightarrow{\mathbf{b}}-5 \overrightarrow{\mathbf{c}}$ intersect
at
- A$-4 \overrightarrow{\mathbf{a}}$
- B$4 \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}}$
- C$4 \overrightarrow{\mathbf{c}}$
- Correct answerDNone of these
Solution
The equations of the lines joining $6 \overrightarrow{\mathbf{a}}-4 \overrightarrow{\mathbf{b}}+4 \overrightarrow{\mathbf{c}},-4 \overrightarrow{\mathbf{c}} \quad$ and
$-\overrightarrow{\mathbf{a}}-2 \overrightarrow{\mathbf{b}}-3 \overrightarrow{\mathbf{c}}, \overrightarrow{\mathbf{a}}+2 \overrightarrow{\mathbf{b}}-5 \overrightarrow{\mathbf{c}}$ are respectively
$$
\overrightarrow{\mathbf{r}}=6 \overrightarrow{\mathbf{a}}-4 \overrightarrow{\mathbf{b}}+4 \overrightarrow{\mathbf{c}}+m(-6 \overrightarrow{\mathbf{a}}+4 \overrightarrow{\mathbf{b}}-8 \overrightarrow{\mathbf{c}}) ...(i)
$$
and $\overrightarrow{\mathbf{r}}=-\overrightarrow{\mathbf{a}}-2 \overrightarrow{\mathbf{b}}-3 \overrightarrow{\mathbf{c}}+n(2 \overrightarrow{\mathbf{a}}+4 \overrightarrow{\mathbf{b}}-2 \overrightarrow{\mathbf{c}})$ ...(ii)
For the point of intersection, the Eqs. (i) and
(ii) should give the same value of $\overrightarrow{\mathbf{r}}$. Hence equating the coefficients of vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ in the two expressions for $\overrightarrow{\mathbf{r}}$, we get
$$
\begin{array}{l}
6 m+2 n=7 \\
2 m-2 n=1
\end{array}
$$
and
$$
8 m-2 n=7
$$
On solving Eqs. (iii) and (iv), we get $m=1, n=\frac{1}{2} .$ These values of $m$ and $n$, also satisfy the Eq. (v).
$\therefore$ The lines intersect. Putting the value of $m$ in Eq. (i), we get the position vector of the point of intersection as $-4 \overrightarrow{\mathbf{c}}$.
Hence, the correct answer is option: 'D'.