MP Board · Class 10 · Mathematics · Coordinate GeometryDetermine the ratio in which the line segment joining the points $(-3, 10)$ and $(6, -8)$ is divided by the point $(-1, 6)$. Also, find the coordinates of the point which divides the join of $(-1, 7)$ and $(4, -3)$ in the ratio $2:3$.
Part 1: Finding the Ratio
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Given points: Let $A(x_1, y_1) = (-3, 10)$ and $B(x_2, y_2) = (6, -8)$. Let the dividing point be $P(x, y) = (-1, 6)$.
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Section Formula: The coordinates of a point $P(x, y)$ dividing the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m_1 : m_2$ are given by: $x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}$ $y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}$
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Assumption for Ratio: Let the ratio be $k : 1$ (i.e., $m_1 = k$ and $m_2 = 1$).
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Applying the Formula for x-coordinate: $-1 = \frac{k(6) + 1(-3)}{k + 1}$ $-1(k + 1) = 6k - 3$ $-k - 1 = 6k - 3$ $-1 + 3 = 6k + k$ $2 = 7k$ $k = \frac{2}{7}$
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Verification with y-coordinate: $6 = \frac{\frac{2}{7}(-8) + 1(10)}{\frac{2}{7} + 1}$ $6 = \frac{-\frac{16}{7} + 10}{\frac{9}{7}}$ $6 = \frac{\frac{-16 + 70}{7}}{\frac{9}{7}}$ $6 = \frac{54}{9} = 6$ Since LHS equals RHS, the ratio is correct.
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Conclusion for Part 1: The required ratio is $2:7$ internally.
Part 2: Finding Coordinates of the Dividing Point
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Given points: $A(x_1, y_1) = (-1, 7)$ $B(x_2, y_2) = (4, -3)$ Ratio $m_1 : m_2 = 2 : 3$
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Applying Section Formula: $x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}$ $x = \frac{2(4) + 3(-1)}{2 + 3}$ $x = \frac{8 - 3}{5} = \frac{5}{5} = 1$
$y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}$ $y = \frac{2(-3) + 3(7)}{2 + 3}$ $y = \frac{-6 + 21}{5} = \frac{15}{5} = 3$
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Conclusion for Part 2: The coordinates of the point dividing the line segment are $(1, 3)$.