Question

Find the ratio in which the line segment joining the points (-2, 4) and (7, 3) is divided by the

Y - axis.​

Answer 1

Given: A line segment joining the points (-2, 4) and (7, 3) divided by y-axis.

To find: Ratio in which line segment is divided by y-axis.

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━

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

• If a point (x, y) divides the line segment joining the points (x₁, y₁) and (x₂, y₂) internally in the ratio m₁ : m₂ are given by the formula:

⠀

$\star\;{\boxed{\sf{\pink{(x,y) = \bigg( \dfrac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\;,\;\dfrac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \bigg) }}}}$

$\bf Here \begin{cases} & \sf{(x_1 , y_1) = (-2 ,4)} \\ & \sf{(x_2 , y_2) = (7, 3)} \end{cases}$

$\dag\;{\underline{\frak{Putting\;values\;on\;formula,}}}$

$:\implies\sf (x,y) = \bigg( \dfrac{m_1 (7) + m_2 (-2)}{m_1 + m_2}\;,\;\dfrac{m_1 (7) + m_2 (4)}{m_1 + m_2} \bigg)\\ \\ :\implies\sf (x,y) = \bigg( \dfrac{7m_1 - 2m_2}{m_1 + m_2}\;,\;\dfrac{7 m_1 + 4 m_2}{m_1 + m_2} \bigg)$

Given points lies on y - axis.

Therefore, x - cordinate = 0.

⠀

$:\implies\sf \dfrac{7m_1 - 2m_2}{m_1 + m_2} = 0\\ \\ :\implies\sf 7m_1 - 2m_2 = 0(m_1 + m_2)\\ \\ :\implies\sf 7m_1 - 2m_2 = 0\\ \\ :\implies\sf 7m_1 = 2m_2\\ \\ :\implies\sf \dfrac{m_1}{m_2} = \dfrac{2}{7}\\ \\ :\implies{\underline{\boxed{\frak{\purple{m_1 : m_2 = 2 : 7}}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;The\;line\; segment\; divided\;by\;y-axis\;in\;ratio\; {\textsf{\textbf{2:7}}}.}}}$

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━

$\qquad\qquad\boxed{\underline{\underline{\pink{\bigstar \: \bf\:More\:to\:know\:\bigstar}}}}$

• Distance Formula: The distance formula is a formula that is used to find the distance between two points. These points can be in any dimension.

• $\sf d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Answer 2

$\star\underline{\mathtt\orange{❥Q} \mathfrak\blue{u }\mathfrak\blue{E} \mathbb\purple{ s}\mathtt\orange{T} \mathbb\pink{iOn}}\star\:$

Find the ratio in which the line segment joining the points (-2, 4) and (7, 3) is divided by the

Y - axis.

$\star\underbrace{\mathtt\red{❥ᴀ} \mathtt\green{n }\mathtt\blue{S} \mathtt\purple{W}\mathtt\orange{e} \mathtt\pink{R}}\star\:$

${ { \underbrace{ \mathbb{ \red{GiVeN\ }}}}}$

$A\: line\: segment\: joining\: the\: points\: (-2, 4)\\ and\: (7, 3) \:divided\: by\: y-axis.$

${ { \underbrace{ \mathbb{ \red{To\:PrOvE\ }}}}}$

$Ratio\: in \:which \:line \:segment\: is\: divided\\ by\: y-axis.$

${ \color{aqua}{ \underbrace{ \underline{ \color{lime}{ \mathbb{\star SoLuTiOn\star }}}}}}$

$Using \:section\: formula$

$\begin{gathered}\star\;{\boxed{\sf{\blue{(x,y) = \bigg( \dfrac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\;,\;\dfrac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \bigg) }}}}\\ \\\end{gathered}$

$\begin{gathered}\bf Here \begin{cases} & \sf{(x_1 , y_1) = (-2 ,4)} \\ & \sf{(x_2 , y_2) = (7, 3)} \end{cases}\\ \\\end{gathered}$

$Substitute\: the \:values$

$\begin{gathered}:\sf (x,y) = \bigg( \dfrac{m_1 (7) + m_2 (-2)}{m_1 + m_2}\;,\;\dfrac{m_1 (7) + m_2 (4)}{m_1 + m_2} \bigg)\\ \\ :\sf (x,y) = \bigg( \dfrac{7m_1 - 2m_2}{m_1 + m_2}\;,\;\dfrac{7 m_1 + 4 m_2}{m_1 + m_2} \bigg)\\ \\\end{gathered}$

$Given \:points \:lies \:on\:y-axis$

$\therefore, x - cordinate = 0$

$\begin{gathered}:\sf \dfrac{7m_1 - 2m_2}{m_1 + m_2} = 0\\ \\ :\sf 7m_1 - 2m_2 = 0(m_1 + m_2)\\ \\ :\sf 7m_1 - 2m_2 = 0\\ \\ :\sf 7m_1 = 2m_2\\ \\ :\sf \dfrac{m_1}{m_2} = \dfrac{2}{7}\\ \\ :{\underline{\boxed{\frak{\blue{m_1 : m_2 = 2 : 7}}}}}\;\bigstar\\ \\\end{gathered}$

$\therefore The\:line\:segment\:divided\:by\:y−axis\:in\:ratio\:{\blue{\boxed {\boxed {2:7.}}} }$

$\blue{\boxed{\blue{ \bold{\fcolorbox{red}{black}{\green{Hope\:It\:Helps}}}}}}$

${\mathbb{\colorbox {orange} {\boxed{\boxed{\boxed{\boxed{\boxed{\colorbox {lime} {\boxed{\boxed{\boxed{\boxed{\boxed{\colorbox {aqua} {@suraj5069}}}}}}}}}}}}}}}$