point C(2,3) divides internally the line segment joining A(3,5) B internally ratio of 1 is to 2
Answers
Answer:
We have given that point C divides the line segment AB in the ratio 1 : 2.
We have to find the coordinates of the point B.
\begin{gathered}\bullet\sf\:A\:\equiv\:(\:3\:,\:5\:)\:\equiv\:(\:x_{1}\:,\:y_{1}\:)\\\\\bullet\sf\:B\:\equiv\:(\:x_{2}\:,\:y_{2}\:)\\\\\bullet\sf\:C\:\equiv\:(\:2\:,\:3\:)\:\equiv\:(\:x\:,\:y\:)\end{gathered}
∙A≡(3,5)≡(x
1
,y
1
)
∙B≡(x
2
,y
2
)
∙C≡(2,3)≡(x,y)
Now, we know that,
\begin{gathered}\pink{\sf\:x\:=\:\dfrac{\:mx_{2}\:+\:nx_{1}}{m\:+\:n}\:\:,\:\:y\:=\:\dfrac{my_{2}\:+\:ny_{1}}{m\:+\:n}}\:\sf\:\:-\:-\:[\:Section\:formula\:]\\\\:\implies\sf\:2\:=\:\dfrac{1\:(\:x_{2}\:)\:+\:2\:(\:3\:)}{1\:+\:2\:}\:\:,\:\:3\:=\:\dfrac{1\:(\:y_{2}\:)\:+\:2\:(\:5\:)}{1\:+\:2\:}\\\\:\implies\sf\:2\:=\:\dfrac{x_{2}\:+\:6}{3}\:\:,\:\:3\:=\:\dfrac{y_{2}\:+\:10}{3}\\\\:\implies\sf\:2\:\times\:3\:=\:x_{2}\:+\:6\:\:,\:\:3\:\times\:3\:=\:y_{2}\:+\:10\\\\:\implies\sf\:6\:=\:x_{2}\:+\:6\:\:,\:\:9\:=\:y_{2}\:+\:10\\\\:\implies\sf\:x_{2}\:=\:6\:-\:6\:\:,\:\:y_{2}\:=\:9\:-\:10\\\\:\implies\boxed{\red{\sf\:x_{2}\:=\:0\:\:,\:\:y_{2}\:=\:-\:1}}\end{gathered}
x=
m+n
mx
2
+nx
1
,y=
m+n
my
2
+ny
1
−−[Sectionformula]
:⟹2=
1+2
1(x
2
)+2(3)
,3=
1+2
1(y
2
)+2(5)
:⟹2=
3
x
2
+6
,3=
3
y
2
+10
:⟹2×3=x
2
+6,3×3=y
2
+10
:⟹6=x
2
+6,9=y
2
+10
:⟹x
2
=6−6,y
2
=9−10
:⟹
x
2
=0,y
2
=−1