Question

Find the ratio in which x axis divides the line segment joining the points 2, - 3 and 5, 6 then find the intersecting point on x-axis

Answer 1

Given: Points (2, -3) and (5, 6) being divided by the x - axis.

To find: The ratio in which it is divided and the intersecting point.

Answer:

Let's assume the ratio is k:1.

Since it's given that it's been divided through the x - axis, the point is  (x, 0).

Section formula:

$\tt \bigg(\dfrac{mx_2\ +\ nx_1}{m\ +\ n},\ \dfrac{my_2\ +\ ny_1}{m\ +\ n}\bigg)$

From the given data, we have:

$\tt m\ =\ k\\\\n\ =\ 1\\\\x_1\ =\ 2\\\\x_2\ =\ 5\\\\y_1\ =\ -3\\\\y_2\ =\ 6$

Using them in the formula,

$\tt (x, 0)\ =\ \bigg(\dfrac{(k\ \times\ 5)\ +\ (1\ \times\ 2)}{k\ +\ 1},\ \dfrac{(k\ \times\ 6)\ +\ (1\ \times\ -3)}{k\ +\ 1}\bigg)\\\\\\(x, 0)\ =\ \bigg(\dfrac{5k\ +\ 2}{k\ +\ 1},\ \dfrac{6k\ -\ 3}{k\ +\ 1}\bigg)$

Equating the y - coordinates,

$\tt 0\ =\ \dfrac{6k\ -\ 3}{k\ +\ 1}\\\\\\0\ =\ 6k\ -\ 3\\\\\\3\ =\ 6k\\\\\\\dfrac{3}{6}\ =\ k\\\\\\\dfrac{1}{2}\ =\ k$

Therefore, the ratio is 1:2.

Since we have the value of k, let's substitute it whilst equating the x - coordinates.

$\tt x\ =\ \dfrac{5k\ +\ 2}{k\ +\ 1}\\\\\\x\ =\ \dfrac{(5\ \times\ \frac{1}{2})\ +\ 2}{\frac{1}{2}\ +\ 1}\\\\\\x\ =\ \dfrac{\frac{5}{2}\ +\ 2}{\frac{1\ +\ 2}{2}}\\\\\\x\ =\ \dfrac{\frac{5\ +\ 4}{2}}{\frac{3}{2}}\\\\\\x\ =\ \dfrac{\frac{9}{2}}{\frac{3}{2}}\\\\\\x\ =\ \dfrac{9}{2}\ \times\ \dfrac{2}{3}\\\\\\x\ =\ 3$

Therefore, (3, 0) divides the line segment joining the points (2, -3) and (5, 6) in the ratio 1:2.

Answer 2

$\huge{\underline{\underline{\red{Answer}}}}$

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$\mathbb{\boxed{\green{GIVEN}}}$

• 1st point (2,-3)
• 2nd point (5,6)
• Intersected by X axis.

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$\bf{\mathbb{\underline{\red{SOLUTION}}}}$

by section formula we know,

if ($x_{1}$$y_{1}$) And ($x_{2}$ $y_{2}$) are co-ordinates of two points, and if they're interested in ratio m:n,

then the point of intersection will be:

$( \frac{m(x2) + n(x1)}{m + n} ; \: \frac{m(y2) + n(y1)}{m + n} )$

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It's given x axis intersects the line joining the points.

Thus the point of intersection will have the coordinate (x,0).

Thus

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Now

Let the ratio be m:n

★$x_{1}$ = 2

★$y_{1}$= -3

★$x_{2}$=5

★$y_{2}$=6

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As we know the value of y interscrept is 0, (as the point is on x axis)

Then putting the values in section formula we get,

$( \frac{m(y2) + n(y1)}{m + n} ) = 0 \\ \implies\frac{m(6) + n( - 3)}{m + n} = 0 \\ \implies 6m - 3n=(m + n) \times 0 \\ \implies6m - 3n = 0 \\ \implies \: 6m = 3n \\ \implies \frac{m}{n} = \frac{3}{6} \\ \implies \: \frac{m}{n} = \frac{1}{2}$

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$\large{\boxed{\pink{Ratio\:is\:1:2}}}$

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thus

m= 1

n=2

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Now,

$X=( \frac{m(x2)+n(x1)}{m+n})\\</p><p>\implies X= (\frac{1(5)+2(2)}{1+2}) \\</p><p>\implies X= \frac{5+4}{3} \\</p><p>\implies X= \frac{9}{3} \\</p><p>\implies X= 3$

${\bold{\underline{\green{x=3}}}}$

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The point is ${\large{\boxed{\pink{Point\:is=\:(3,0)}}}}$