In what ratio does the X − axis divide the line segment joining the points − 4 − 6 and − 1 7 also find the coordinates of the point of division?

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Hint: The section formula for the point (x, y) which divides the line segment joining the points \[({x_1},{y_1})\] and \[({x_2},{y_2})\] in the ratio m:n is given as \[(x,y) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\]. Use this formula to find the ratio in which the x-axis divides the line segment joining the points (1, 2) and (2, 3).Complete step-by-step answer:We need to find the ratio in which the x-axis divides the line segment joining the points (1, 2) and (2, 3).Any point on the x-axis has its y coordinate equal to zero. Hence, let us assume a point (x, 0) that divides the line segment joining the points (1, 2) and (2, 3).The section formula for the point (x, y) which divides the line segment joining the points \[({x_1},{y_1})\] and \[({x_2},{y_2})\] in the ratio m:n is given as follows:\[(x,y) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\]

Let the ratio in which the point (x, 0) divides the line segment joining the points (1, 2) and (2, 3) be m:n. then we have:\[(x,0) = \left( {\dfrac{{m(2) + n(1)}}{{m + n}},\dfrac{{m(3) + n(2)}}{{m + n}}} \right)\]Simplifying, we have:\[(x,0) = \left( {\dfrac{{2m + n}}{{m + n}},\dfrac{{3m + 2n}}{{m + n}}} \right)\]The corresponding coordinates are equal. Equating the y-coordinates, we have:\[\dfrac{{3m + 2n}}{{m + n}} = 0\]Simplifying, we have:\[3m + 2n = 0\]\[3m = - 2n\]Finding the ratio of m and n, we have:\[\dfrac{m}{n} = \dfrac{{ - 2}}{3}\]Hence, the x-axis divides the line segment joining the points (1, 2) and (2, 3) in the ratio – 2:3.Hence, the correct answer is option (d).Note: You can also find the equation of the line segment joining the points (1, 2) and (2, 3) and then find the point of intersection with the x-axis. Then use distance between two points formula \[\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} \] to find the ratio.

The ratio in which the y-axis divides two points (x1 , y1) and (x2 , y2) is \[\lambda: 1\]

The co-ordinates of the point dividing two points (x1 , y1) and (x2 , y2) in the ratio m : n is given as,

`(x , y) = ((lambdax_2 + x_1)/(lambda + 1 )) ,((lambday_2 + y_1)/(lamda + 1))` where, `lambda = m/n`

Here the two given points are A(5,−6) and B(−1,−4).

\[(x, y) = \left( \frac{- \lambda + 5}{\lambda + 1}, \frac{- 4\lambda - 6}{\lambda + 1} \right)\]

Since, the y-axis divided the given line, so the x coordinate will be 0.

\[\frac{- \lambda + 5}{\lambda + 1} = 0\]
\[\lambda = \frac{5}{1}\]

Thus the given points are divided by the y-axis in the ratio 5:1.

The co-ordinates of this point (x, y) can be found by using the earlier mentioned formula.

`(x , y ) = ((5/1 (-1) + (5) )/(5/1 + 1)) , ((5/1(-4)+(-6))/(5/1 +1))`

`(x , y) = (0/6) , (-26/6)`

`(x , y ) = ( 0 , - 26/6)`

Thus the co-ordinates of the point which divides the given points in the required ratio are `(0,-26/6)`.

In what ratio does the x axis divide the line segment joining the points 4, 6 and 1,7? Find the coordinates of the point of division.

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In what ratio, does the X axis divides the line segment joining the points 2, 3 and 5, 6? [4 MARKS]