Find the ratio in which the line segment joining the points A(6,4) and B(1,-7) is divided by the x axis . Also find the distance 2AB and the coordinates of the point on the x axis which cuts the line segment joining A and B in the required ratio

Let the coordinate of the point at x – axis be (x, 0).

Let the ratio be m : n.

Applying section formula,

Putting (x ₁, y ₁) = (6,4), (x ₂, y ₂) = (1,-7) and (x, y) = (x, 0)
$$\begin{gathered} \mathrm{x}=\frac{1\mathrm{m}+6\mathrm{n}}{\mathrm{m}+\mathrm{n}},0=\frac{-7\mathrm{m}+4\mathrm{n}}{\mathrm{m}+\mathrm{n}} \\ \mathrm{Taking} 0=\frac{-7\mathrm{m}+4\mathrm{n}}{\mathrm{m}+\mathrm{n}} \\ \Rightarrow -7\mathrm{m}+4\mathrm{n}=0 \\ \Rightarrow 7\mathrm{m}=4\mathrm{n} \\ \Rightarrow \frac{\mathrm{m}}{\mathrm{n}}=\frac{4}{7} \\ \mathrm{So}, \mathrm{m}:\mathrm{n}=4:7 \\ \mathrm{Thus}, \mathrm{line} \mathrm{segment} \mathrm{joining} (6,4) \mathrm{and} (1,-7) \mathrm{is} \mathrm{divided} \mathrm{by} \mathrm{x}-\mathrm{axis} \mathrm{in} \mathrm{the} \mathrm{ratio} 4:7 \end{gathered}$$
$\mathrm{Distance} 2\mathrm{AB}=2\sqrt{{\left(1-6\right)}^2+(-7-4{)}^2}=2\sqrt{5^2+{11}^2}=2\sqrt{25+121}=2\sqrt{146}$
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