Question

4. In what ratio does the point (1, a) divide the
join of (-1, 4) and (4, -1) ?
Also, find the value of a.​

Answer 1

${\bold{\underline{\underline{Answer:}}}}$

${\bold{\therefore Ratio=2:3}}$

${\bold{\therefore a=2}}$

${\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

• In the given question information given about a line segment in which point P divides the line in ratio.

• We have to find the ratio in which P divides the line and value of a.

$\underline \bold{Given : } \\ \implies \bold{Let \: AB \: be \: the \: line \: and \: P \: divides \: it} \\ \\ \implies coordinate \: of \: A = ( - 1,4) \\ \\ \implies coordinate \: of \: B = ( 4, - 1) \\ \\ \implies coordinate \: of \: P = ( 1,a) \\ \\ \underline \bold{to \: find : } \\ \implies Value \:of \: a = ? \\ \\ \implies Ratio = ?$

• According to given question :

$\bold{Let \: P\: divides \: in \: \:ratio \: = k : 1} \\ \\ \bold{By \: section \: formula : } \\ \implies x = \frac{ mx_{2} + nx_{1} }{m + n} \\ \\ \implies 1 = \frac{k \times 4 + 1 \times - 1}{k + 1} \\ \\ \implies k + 1 = 4k - 1 \\ \\ \implies k - 4k = - 1 - 1 \\ \\ \implies \cancel{-} 3k = \cancel{-} 2 \\ \\ \bold{\implies k = \frac{2}{3}} \\ \\ \bold{ \therefore Ratio = 2 : 3} \\ \\ \bold{For \: y \: ordinate : } \\ \implies y = \frac{ my_{2} + ny_{1} }{m + n} \\ \\ \implies a = \frac{2 \times - 1 + 3 \times 4}{2 + 3} \\ \\ \implies 5a = - 2 + 12 \\ \\ \implies a = \frac{\cancel{10}}{\cancel5} \\ \\ \bold{\implies a = 2}$

Answer 2

Solution:

Given:

=> A = (-1, 4)

=> B = (4, -1)

=> P = (1, a)

To Find:

=> Ratio

=> Value of a

Formula used:

$\sf{\implies \Bigg[\dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}},\;\dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\Bigg]\;\;\;\;\;\;(Section\;formula)}$

So,

Let p divides in the ratio k : 1. Then,

For x coordinate:

$\sf{\implies x = \dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}}$

$\sf{\implies 1 = \dfrac{k\times 4+1\times (-1)}{k+1}}$

$\sf{\implies k+1=4k-1}$

$\sf{\implies k - 4k = -1-1}$

$\sf{\implies -3k=-2}$

$\large{\boxed{\boxed{\sf{\implies k = \frac{3}{2}}}}}$

For y coordinate:

$\sf{\implies y = \dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}}$

$\sf{\implies a = \dfrac{2\times (-1)+3\times 4}{2+3}}$

$\sf{\implies 5a = 10}$

$\sf{\implies a = \frac{10}{5}}$

$\large{\boxed{\boxed{\sf{\implies a = 2}}}}$