Question

Find the ratio in which the line joining (3,4) and (-4,7) is divided by y-axis. Also find the co-ordinates of the point of intersection.​

Answer 1

GiveN :-

• Two points (3,4) and (-4,7) is given and a line is drawn through it passing through the y axis.

To FinD :-

• Thr ratio in which it is divided by y axis.
• The co-ordinates of point of intersection.

SolutioN :-

Here the two points given to us are (3,4) and (-4,7) . We know that at y axis , x coordinate is 0 . Let us take the y coordinate be y . Then the point through which the line passes at y axis is (0,y).

Let us take that,

$\red{\frak{Let }}\begin{cases}\textsf{ Coordinates at y axis = \textbf{(0,y)} .} \\\textsf{ The ratio in which it divides = \textbf{k:1}.}\end{cases}$

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G R A P H :-

$\setlength{\unitlength}{1 cm}\begin{picture}(12,8)\thicklines \put(1,1){\vector(1,0){5}} \put(1,1){\vector(0,1){5}}\put(4,.5){\line(-1,1){3.2}}\multiput(1.2,1.4)(0,0.5){5}{\line(-1,0){0.4}}\put(1,3.5){\circle*{0.15}}\put(1.3,3.5){ \sf (0,5.28) }\multiput(1.4,.8)(.5,0){8}{\line(0,1){0.4}}\put(3,5){ \boxed{\sf @ RishabhRanjan }} }\end{picture}$

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Now here we can use the Section Formula ,

$\sf:\implies \pink{(x,y) = \bigg(\dfrac{m_2x_1+m_1x_2}{m_1+m_2} , \dfrac{m_2y_1+m_1y_2}{m_2+m_1}\bigg) }\qquad\bigg\lgroup \red{\tt Section\ Formula }\bigg\rgroup \\\\\sf:\implies (0,y) = \dfrac{(k)(-4)+(1)(3)}{k+1} , \dfrac{(k)(7) + (1)(4) }{k+1} \\\\\sf:\implies (0,y) = \dfrac{-4k+3}{k+1} , \dfrac{7k+4}{k+1} \\\\ \qquad\qquad\tiny{\sf( \dag\:\:\:\:\red{On \ Comparing \ x - coordinate } )}\\\\\sf:\implies 0 = \dfrac{-4k+3}{k+1} \\\\\sf:\implies -4k+3 = 0 \\\\\sf:\implies -4k = -3 \\\\\sf:\implies k =\dfrac{-4}{-3} \\\\\sf:\implies\underset{\blue{\sf Required \ Ratio } }{\underbrace{\boxed{\pink{\frak{ k:1 = 3:4 }}}}}$

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$\red{\bigstar}\underline{\textsf{ Finding the coordinates of point . }}$

$\sf:\implies\pink{ y = \dfrac{7k+4}{k+1} }\\\\\sf:\implies y = \dfrac{7\bigg(\dfrac{3}{4}\bigg) + 4 }{\dfrac{3}{4}+1} \\\\\sf:\implies y = \dfrac{ \dfrac{21}{4}+4}{0.75+1}\\\\\sf:\implies y = \dfrac{5.25 + 4 }{1.75} \\\\\sf:\implies y = \dfrac{9.25}{1.75} \\\\\sf:\implies \underset{\blue{\sf Y - coordinate } }{\underbrace{\boxed{\pink{\frak{ y = 5.28 }}}}}$

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$\red{\bigstar}\boxed{\orange{\tt Required\ coordinate = (0,5.28 ) }}$

$\red{\bigstar}\boxed{\orange{\tt Required\ Ratio = (3,4) }}$

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