Question

Find the ratios in which the line segment joining (-2,-3) and (5,4) is divided by the co-ordinates axes and hence find the co-ordinates of these points.​

Answer 1

Let the points on the axes which divide the line segment joining (-2, -3) and (5, 4) be (x, 0) and (0, y), which are on x axis and y axis respectively.

Then, by two point form,

$\displaystyle\longrightarrow\sf{\dfrac{0-4}{x-5}=\dfrac{-3-4}{-2-5}}$

$\displaystyle\longrightarrow\sf{\dfrac{-4}{x-5}=1}$

$\displaystyle\longrightarrow\sf{x=1}$

And,

$\displaystyle\longrightarrow\sf{\dfrac{y-4}{0-5}=1}$

$\displaystyle\longrightarrow\sf{y=-1}$

Hence the points are (1, 0) and (0, -1).

Let the point (0, -1) divide the line in the ratio m : n in the order from left to right. Then, by section formula,

$\displaystyle\longrightarrow\sf{0=\dfrac{5m-2n}{m+n}}$

$\displaystyle\longrightarrow\sf{\underline{\underline{m:n=2:5}}}$

Hence it divides in the ratio 2:5.

Let the point (1, 0) divide the line in the ratio p : q in the order from left to right. Then, again by section formula,

$\displaystyle\longrightarrow\sf{0=\dfrac{4p-3q}{p+q}}$

$\displaystyle\longrightarrow\sf{\underline{\underline{p:q=3:4}}}$

Hence it divides in the ratio 3:4.

Answer 2

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