If the point (3,4,5) divides the line segment | vedclass

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(B) Let the points be $P(1,2,3)$ and $R(4,5,6)$. The point $Q(3,4,5)$ divides $PR$ in the ratio $\lambda: 1$.Using the section formula,the coordinates

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$(5,6,7)$

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(B) Let the points be $P(1,2,3)$ and $R(4,5,6)$. The point $Q(3,4,5)$ divides $PR$ in the ratio $\lambda: 1$.Using the section formula,the coordinates of $Q$ are given by:$Q = \left( \frac{\lambda(4) + 1(1)}{\lambda + 1}, \frac{\lambda(5) + 1(2)}{\lambda + 1}, \frac{\lambda(6) + 1(3)}{\lambda + 1} \right) = (3,4,5)$Comparing the $x$-coordinates:$\frac{4\lambda + 1}{\lambda + 1} = 3$$4\lambda + 1 = 3\lambda + 3$$\lambda = 2$Now,we need to find the point that divides the line segment joining $Q(3,4,5)$ and $P(1,2,3)$ in the ratio $-1: \lambda$ (which is $-1: 2$).Let the required point be $S(x, y, z)$. Using the section formula for internal division with ratio $m:n = -1: 2$:$x = \frac{-1(1) + 2(3)}{-1 + 2} = \frac{-1 + 6}{1} = 5$$y = \frac{-1(2) + 2(4)}{-1 + 2} = \frac{-2 + 8}{1} = 6$$z = \frac{-1(3) + 2(5)}{-1 + 2} = \frac{-3 + 10}{1} = 7$Thus,the required point is $(5,6,7)$.

If the point $(3,4,5)$ divides the line segment joining the points $(1,2,3)$ and $(4,5,6)$ in the ratio $\lambda: 1$,then the point which divides the line segment joining the points $(3,4,5)$ and $(1,2,3)$ in the ratio $-1: \lambda$ is

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