Question

POS
tag?
4. Find the ratio in which the line segment joining the points (-3, 10) and (6,- 8) is divided
by (-1,6).​

Answer 1

Appropriate Question:

• Find the ratio in which the line segment joining the points (-3, 10) and (6,- 8) is divided by (-1,6).

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❒ DIAGRAM:

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

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❒ Let the given points be A(-3,10), B(6,-8) and C(-1,6).

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Given that,

• The point C(-1, 6) divide internally the line segment joining points A(-3,10) and B(6,-8) in the ratio.

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

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$\dag\:\boxed{\sf{\pink{x = \dfrac{m_{1}x_{2} +m_{2}x_{1}}{m_{1} + m_{2}}}}}$

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$\underline{\bf{\dag} \:\mathfrak{Substituting\: values \; now\: :}}$

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$:\implies\sf -1 = \dfrac{6k -3}{k + 1} \\\\\\:\implies\sf -1 \Big(k + 1 \Big) = 6k - 3 \\\\\\:\implies\sf -k -1 = 6k - 3 \\\\\\:\implies\sf -k -6k = -3 + 1 \\\\\\:\implies\sf -7k = -2 \\\\\\:\implies\sf k = \dfrac{-7}{-2} \\\\\\:\implies{\underline{\boxed{\frak{\purple{k = \dfrac{7}{2}}}}}}\:\bigstar$

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$\therefore\:{\underline{\sf{Hence, \ the \ required \ ratio \: \ is\: \bf{2:7}.}}}$

Answer 2

$\Large{\orange{\underline{\textsf{\textbf{QUESTION\::}}}}}$

Find the ratio in which the line segment joining the points (-3 , 10) and (6 , - 8) is divided by (-1 , 6).

$\Large{\green{\underline{\textsf{\textbf{Step-by-step\:Explanations\::}}}}}$

Gɪᴠᴇɴ :

• The points (-3 , 10) & (6 , -8) of a line segment is divided by a point (-1 , 6)

Lᴇᴛ,

As sʜᴏᴡɴ ɪɴ ᴛʜᴇ ᴀᴛᴛᴀᴄʜᴍᴇɴᴛ ᴅɪᴀɢʀᴀᴍ,

1. P(-3 , 10)
2. Q(-1 , 6)
3. O(6 , -8)

Tᴏ Fɪɴᴅ :

• The ratio in which the line segments is joining the points P & Q.

Cᴀʟᴄᴜʟᴀᴛɪᴏɴ :

Lᴇᴛ,

• The ratio be k : 1.

Hᴇɴᴄᴇ,

• m₁ = k

• m₂ = 1

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

➣ Section formula is

$\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{x\:=\:\dfrac{m_1\:x_1\:+\:m_2\:x_2}{m_1\:+\:m_2}\:}}}}}}$

Wʜᴇʀᴇ,

• x₁ = -3

• y₁ = 10

• x₂ = 6

• y₂ = -8

• x = -1

• y = 6

$:\implies\:\:\rm{-1\:=\:\dfrac{k\times{6}\:+\:1\times{-3}}{k\:+\:1}\:}$

$:\implies\:\:\rm{-1\:=\:\dfrac{6k\:-\:3}{k\:+\:1}\:}$

$:\implies\:\:\rm{6k\:-\:3\:=\:-(k\:+\:1)\:}$

$:\implies\:\:\rm{6k\:-\:3\:=\:-k\:-\:1\:}$

$:\implies\:\:\rm{6k\:+\:k\:=\:3\:-\:1\:}$

$:\implies\:\:\rm{7k\:=\:2\:}$

$:\implies\:\:\rm{k\:=\:\dfrac{2}{7}\:}$

Hᴇɴᴄᴇ,

➛ The ratio is k : 1 = $\bf{\dfrac{2}{7}\::\:1}$

➣ Multiple 7 on both side, we get

➛ $\rm{\Big(\dfrac{2}{7}\times{7}\Big)\::\:(1\times{7})}$

➛ $\bf\pink{2\::\:7}$

$\Large\bf\purple{Therefore,}$

The ratio in which the line segment joining the points (-3, 10) & (6 ,- 8) is 2 : 7.