If (2, 3, c) are the direction ratios of a ra | vedclass

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(B) The midpoint $M$ of the line segment joining $A(p, -4, 2)$ and $B(3, 2, -4)$ is given by $M = \left( \frac{p+3}{2}, \frac{-4+2}{2}, \frac{2-4}{2}

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$34$

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(B) The midpoint $M$ of the line segment joining $A(p, -4, 2)$ and $B(3, 2, -4)$ is given by $M = \left( \frac{p+3}{2}, \frac{-4+2}{2}, \frac{2-4}{2} \right) = \left( \frac{p+3}{2}, -1, -1 \right)$.Since the ray passes through $C(5, q, 1)$ and $M\left( \frac{p+3}{2}, -1, -1 \right)$,its direction ratios are proportional to the differences of the coordinates: $\left( \frac{p+3}{2} - 5, -1 - q, -1 - 1 \right)$.Given the direction ratios are $(2, 3, c)$,we equate them:$1) \frac{p+3}{2} - 5 = 2 \implies \frac{p+3}{2} = 7 \implies p+3 = 14 \implies p = 11$.$2) -1 - q = 3 \implies q = -4$.$3) -1 - 1 = c \implies c = -2$.Finally,we calculate $c \cdot (p + 7q) = -2 \cdot (11 + 7(-4)) = -2 \cdot (11 - 28) = -2 \cdot (-17) = 34$.

If $(2, 3, c)$ are the direction ratios of a ray passing through the point $C(5, q, 1)$ and also the midpoint of the line segment joining the points $A(p, -4, 2)$ and $B(3, 2, -4)$,then $c \cdot (p + 7q) = $

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