on what ratio does the point (-4,6) divide the line segment joining the points A(-6,10) and B(3,-8)?
Answers
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Answer:
{\large{\bold{\bf{\sf{\underline{Understanding \: the \: question}}}}}}
Understandingthequestion
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}}}}}}
Diagramofthisquestion
\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}}}}}}
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}}}}}}
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}}}}}}
Usingconcept
Section formula
{\large{\bold{\bf{\sf{\underline{Using \: formula}}}}}}
Usingformula
\begin{gathered}\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}\end{gathered}
★
(x,y)=(
m+n
mx
2
+nx
1
,
m+n
my
2
+ny
1
)
{\large{\bold{\bf{\sf{\underline{Full \: Solution}}}}}}
FullSolution
~ Using section formula
\begin{gathered}\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}\end{gathered}
★
(x,y)=(
m+n
mx
2
+nx
1
,
m+n
my
2
+ny
1
)
~ Here the values are
x₁ , y₁ = -6,10
x₂ , y₂ = 3,-8
~ Putting the values,
⠀
\begin{gathered}\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}\end{gathered}
★
(x,y)=(
m+n
mx
2
+nx
1
,
m+n
my
2
+ny
1
)
\begin{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}\end{gathered}
:⟼
m+n
m×3+n×−6
=−4
:⟼m×3+n×−6=−4m−4n
:⟼3m−6n=−4m−4n
:⟼3m+4m=6n−4n
:⟼7m=2n
:⟼
n
m
=
7
2
:⟼
m:n=2:7
★
{\bold{\green{\frak{Henceforth, \: the \: ratio \: is \: 2:7 \: in \: which \: the \: line \: segment \: join \: the \: point}}}}Henceforth,theratiois2:7inwhichthelinesegmentjointhepoint
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