Find the coordinates of the points of trisection of the line segment joining the points $A(2, -2)$ and $B(-7, 4)$. Also, find the ratio in which the line segment joining these points is divided by the y-axis.

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To find the coordinates of the points of trisection and the ratio of division by the y-axis, let us solve the problem step by step using the section formula from Coordinate Geometry. Step 1: Points of Trisection
Points of trisection divide a line segment into three equal parts. Let the given points be $A(x1, y1) = (2, -2)$ and $B(x2, y2) = (-7, 4)$. Let $P$ and $Q$ be the points of trisection such that $AP = PQ = QB$. Finding point $P$: Point $P$ divides the line segment $AB$ in the ratio $1:2$ (since $AP:PB = 1:2$). Using the section formula: $$\left(x, yight) = \left(\frac{m1x2 + m2x1}{m1 + m2}, \frac{m1y2 + m2y1}{m1 + m2}ight)$| Substituting $m1 = 1$, $m2 = 2$, $x1 = 2$, $y1 = -2$, $x2 = -7$, $y2 = 4$: $$x = \frac{1(-7) + 2(2)}{1 + 2} = \frac{-7 + 4}{3} = \frac{-3}{3} = -1$$ $$y = \frac{1(4) + 2(-2)}{1 + 2} = \frac{4 - 4}{3} = \frac{0}{3} = 0$$ So, the coordinates of $P$ are $(-1, 0)$. Finding point $Q$: Point $Q$ divides the line segment $AB$ in the ratio $2:1$ (since $AQ:QB = 2:1$). Using the section formula with $m1 = 2$, $m2 = 1$: $$x = \frac{2(-7) + 1(2)}{2 + 1} = \frac{-14 + 2}{3} = \frac{-12}{3} = -4$$ $$y = \frac{2(4) + 1(-2)}{2 + 1} = \frac{8 - 2}{3} = \frac{6}{3} = 2$$ So, the coordinates of $Q$ are $(-4, 2)$. Step 2: Ratio in which the line segment is divided by the y-axis
Let the y-axis divide the line segment $AB$ in the ratio $k:1$ at point $R$. Since any point on the y-axis has an x-coordinate of $0$, the coordinates of the point of division $R$ are $(0, y)$.
Using the section formula for the x-coordinate: $$0 = \frac{k(-7) + 1(2)}{k + 1}$| $$0 = -7k + 2$$ $$7k = 2 \implies k = \frac{2}{7}$|
Thus, the required ratio is $k:1 = \frac{2}{7}:1$, which simplifies to $2:7$. Final Answer: The coordinates of the points of trisection are $(-1, 0)$ and $(-4, 2)$. The ratio in which the line segment is divided by the y-axis is $2:7$.

MP Board · Class 10 · Mathematics · Coordinate GeometryFind the coordinates of the points of trisection of the line segment joining the points $A(2, -2)$ and $B(-7, 4)$. Also, find the ratio in which the line segment joining these points is divided by the y-axis.

Step 1: Points of Trisection\nPoints of trisection divide a line segment into three equal parts. Let the given points be $A(x_1, y_1) = (2, -2)$ and $B(x_2, y_2) = (-7, 4)$. Let $P$ and $Q$ be the points of trisection such that $AP = PQ = QB$.