Show that the four points P- Q- R- S with position vectors p - q - r - s - respectively such that 5 p -2 q -6 r -9 s -0 - are coplanar- Also- findthe position vector of the point of intersection of the line segments P R and Q S-

Let the point of intersection of the line segments nbsp;PR nbsp;and nbsp;QS nbsp;is A. Then nbsp; nbsp; nbsp; nbsp;5p→ 2q→+6r→ 9s→ nbsp;=0→.⇒ ...

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Standard XII

Mathematics

Intersection of a Line and Finding Roots of a Parabola

Show that the...

Question

Show that the four points P, Q, R, S with position vectors $\vec{p}$ , $\vec{q}$ , $\vec{r}$ , $\vec{s}$ respectively such that 5 $\vec{p}$ −2 $\vec{q}$ +6 $\vec{r}$ −9 $\vec{s}$ = $\vec{0}$ , are coplanar. Also, find the position vector of the point of intersection of the line segments PR and QS.

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Solution

Let the point of intersection of the line segments $P R$ and $Q S$ is A. Then
$5 \vec{p} - 2 \vec{q} + 6 \vec{r} - 9 \vec{s} = \vec{0} . \Rightarrow 5 \vec{p} + 6 \vec{r} = 2 \vec{q} + 9 \vec{s} the sum of the coefficients on both the sides of the above equation is 11 . So, we divide the given equation with 11 . \Rightarrow \frac{5 \vec{p} + 6 \vec{r}}{11} = \frac{2 \vec{q} + 9 \vec{s}}{11}$

$\frac{5 \vec{p} + 6 \vec{r}}{5 + 6} = \frac{2 \vec{q} + 9 \vec{s}}{2 + 9}$

Therefore, A divides $P R$ in the ratio of $5 : 6$ and $Q S$ in the ratio of $2 : 9$ .
The position vector of the point of intersection of the line segment is $\frac{5 \vec{p} + 6 \vec{r}}{11}, \frac{2 \vec{q} + 9 \vec{s}}{11}$ .

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Show that the four points P, Q, R, S with position vectors p→, q→, r→, s→ respectively such that 5p→−2q→+6r→−9s→=0→, are coplanar. Also, find the position vector of the point of intersection of the line segments PR and QS.