Question

In what ratio, does the point (-4,6) divide the line segment joining the points A(-6,10) and B(3,-8)?​

Answer 1

☯ Let the points A(-6,10) and B(3,-8) is divided by point (-4,6) in the ratio m : n.

$\star$ DIAGRAM

⠀⠀$\setlength{\unitlength}{14mm}\begin{picture}(7,5)(0,0)\thicklines\put(0,0){\line(1,0){5}}\put(5.1, - 0.3){\sf B}\put( - 0.2, - 0.3){\sf A}\put(5.2, 0){\sf (3,-8)}\put( - 0.7, 0){\sf (-6,10)}\put(2.3, 0.2){\sf C}\put(2.2, - 0.3){\sf (-4,6)}\put(5, 0){\circle*{0.1}}\put(2.4, 0){\circle*{0.1}}\put(0, 0){\circle*{0.1}}\put(1,0.2){\sf m}\put(3.5, 0.2){\sf n}\end{picture}$⠀

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$\underline{\bigstar\:\boldsymbol{Using\:section\:formula\::}}$

$\star\;{\boxed{\sf{\pink{(x,y) = \bigg( \dfrac{m x_2 + n x_1}{m + n}\;,\; \dfrac{m y_2 + n y_1}{m + n} \bigg)}}}}$

$\sf Here \begin{cases} \sf{x_1 , y_1 = -6,10} \\ \sf{x_2 , y_2 = 3,-8} \end{cases}$

Therefore,⠀⠀

$:\implies\sf \dfrac{m \times 3 + n \times -6}{m + n} = -4 \\\\\\:\implies\sf m \times 3 + n \times -6 = - 4m -4n\\\\\\:\implies\sf 3m - 6n = -4m - 4n\\\\\\:\implies\sf 3m + 4m = 6n - 4n \\\\\\:\implies\sf 7m = 2n\\\\\\:\implies\sf \dfrac{m}{n} = \dfrac{ 2}{7}\\\\\\:\implies{\underline{\boxed{\sf{\purple{m : n = 2 : 7}}}}}\;\bigstar$

$\therefore\;{\underline{\sf{The\;ratio\; in \;which \;(-4,6)\; divides\; the \;line\; segment\; is\; {\textsf{\textbf{2 : 7}}}.}}}$

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$\qquad\qquad\boxed{\bf{\mid{\overline{\underline{\pink{\bigstar\: Additional \ Information:}}}}}\mid}$

Another formula from this chapter is, Distance formula which is used to find the distance b/w two given points. $\: \: \:\sf D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Answer 2

${\large{\bold{\bf{\sf{\underline{Understanding \: the \: question}}}}}}$

This question says that the point (-4,6) divide the line segment joining the points A(-6,10) and B(3,-8) and we have to find its ratio !

${\large{\bold{\bf{\sf{\underline{Diagram \: of \: this \: question}}}}}}$

$\setlength{\unitlength}{14mm}\begin{picture}(7,5)(0,0)\thicklines\put(0,0){\line(1,0){5}}\put(5.1, - 0.3){\sf B}\put( - 0.2, - 0.3){\bf A}\put(5.2, 0){\bf (3 , -8)}\put( - 0.7, 0){\bf (-6 , 10)}\put(2.3, 0.2){\bf C}\put(2.2, - 0.3){\bf (-4 , 6)}\put(5, 0){\circle*{0.1}}\put(2.4, 0){\circle*{0.1}}\put(0, 0){\circle*{0.1}}\put(1,0.2){\bf m}\put(3.5, 0.2){\bf n}\end{picture}$ ⠀

${\large{\bold{\bf{\sf{\underline{Solution}}}}}}$

2:7 is the ratio which the point (-4,6) divide the line segment joining the points A(-6,10) and B(3,-8)

${\large{\bold{\bf{\sf{\underline{Assumption}}}}}}$

Let the points A(-6,10) and B(3,-8) is divided by line segment in the ratio of m:n ( according to the question and the formula )

${\large{\bold{\bf{\sf{\underline{Using \: concept}}}}}}$

Section formula

${\large{\bold{\bf{\sf{\underline{Using \: formula}}}}}}$

$\begin{gathered}\bigstar\:{\boxed{\bf{\red{(x , y) = \bigg( \dfrac{m x_2 + n x_1}{m + n}\: ,\: \dfrac{m y_2 + n y_1}{m + n} \bigg)}}}}\\ \\\end{gathered}$

${\large{\bold{\bf{\sf{\underline{Full \: Solution}}}}}}$

~ Using section formula

$\begin{gathered}\bigstar\:{\boxed{\bf{\red{(x , y) = \bigg( \dfrac{m x_2 + n x_1}{m + n}\: ,\: \dfrac{m y_2 + n y_1}{m + n} \bigg)}}}}\\ \\\end{gathered}$

~ Here the values are

x₁ , y₁ = -6,10

x₂ , y₂ = 3,-8

~ Putting the values,

⠀

$\begin{gathered}\bigstar\:{\boxed{\bf{(x , y) = \bigg( \dfrac{m x_2 + n x_1}{m + n}\: ,\: \dfrac{m y_2 + n y_1}{m + n} \bigg)}}}\\ \\\end{gathered}$

$\begin{gathered}:\longmapsto\bf \dfrac{m \times 3 + n \times -6}{m + n} = -4 \\\\\\:\longmapsto\bf m \times 3 + n \times -6 = - 4m -4n\\\\\\:\longmapsto\bf 3m - 6n = -4m - 4n\\\\\\:\longmapsto\bf 3m + 4m = 6n - 4n \\\\\\:\longmapsto\bf 7m = 2n\\\\\\:\longmapsto\bf \dfrac{m}{n} = \dfrac{ 2}{7}\\\\\\:\longmapsto{\boxed{\bf{\pink{m : n = 2 : 7}}}}\:\bigstar\\ \\\end{gathered}$

${\bold{\green{\frak{Henceforth, \: the \: ratio \: is \: 2:7 \: in \: which \: the \: line \: segment \: join \: the \: point}}}}$

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${\large{\bold{\bf{\sf{\underline{Additional \: information}}}}}}$

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Formulas of Statistics –

$\boxed {\begin{minipage}{9.2 cm}\\ \dag \: \underline{\Large\bf Formulas\:of\:Statistics} \\ \\ \bigstar \: \underline{\rm Mean:} \\ \\ \bullet\sf M=\dfrac {\Sigma x}{n} \\ \bullet\sf M=a+\dfrac {\Sigma fy}{\Sigma f} \\ \\ \bullet\sf M=A +\dfrac {\Sigma fy^i}{\Sigma f}\times c \\ \\ \bigstar \: \underline{\rm Median :} \\ \\ \bullet\sf M_d=\dfrac {n+1}{2} \:\left[\because n\:is\:odd\:number\right] \\ \bullet\sf M_d=\dfrac {1}{2}\left (\dfrac {n}{2}+\dfrac {n}{2}+1\right)\:\left[\because n\:is\:even\:number\right] \\ \\ \bullet\sf M_d=l+\dfrac {m-c}{f}\times i \\ \\ \bigstar \: {\boxed{\sf M_0=3M_d-2M}}\end {minipage}}$

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Request : Please see this answer from web browser or chrome just saying because I give some formulas and diagram here but they are not shown in app