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(A) Let the $Y$-axis divide the line segment $AB$ at point $P(0, y)$ in the ratio $k:1$.Using the section formula,the coordinates of point $P$ are giv

Vedclass · Accepted Answer

$2:1, (0, 2)$

Vedclass · Answer

(A) Let the $Y$-axis divide the line segment $AB$ at point $P(0, y)$ in the ratio $k:1$.Using the section formula,the coordinates of point $P$ are given by:$P = \left( \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \right)$Here,$m_1 = k$,$m_2 = 1$,$x_1 = -2$,$y_1 = 4$,$x_2 = 1$,$y_2 = 1$.Since the point lies on the $Y$-axis,its $x$-coordinate is $0$:$0 = \frac{k(1) + 1(-2)}{k + 1}$$0 = k - 2 \implies k = 2$.So,the ratio is $2:1$.Now,find the $y$-coordinate:$y = \frac{2(1) + 1(4)}{2 + 1} = \frac{2 + 4}{3} = \frac{6}{3} = 2$.Thus,the point of division is $(0, 2)$ and the ratio is $2:1$.

In which ratio and at which point does the $Y$-axis divide the line segment joining $A (-2, 4)$ and $B (1, 1)$ from $A$?

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