Question

find the ratio in which p(-4,6) divides the line joining the points A(-6,10) andb(3,-8)​

Answer 1

$\large\bf\underline \orange{Given:-}$

• p(-4,6) divides the line segment joining the points A(-6,10) and B(3,-8)

$\large\bf\underline \orange{To \: find:-}$

• Ratio in which p(-4,6) divides the line joining the points A(-6,10) andb(3,-8).

$\huge\bf\underline \green{Solution:-}$

• Let the ratio be k:1

Then,

• By the section formula,

$\bold{ \underline{ \pink { \boxed {\bf{ \mid{\frac{m_1x_2+m_2x_1}{k + 1} \: \: \: \frac{,m_1x_2+m_2x_1}{k + 1 } \mid \: \: \: \: }} }}}}$

• Let m1 = k and m2 = 1

$\rm \: p( \frac{k \times 3 + 1 \times - 6}{k + 1} \: , \: \frac{k \times - 8 + 1 \times 10}{k + 1})$

1st case:-

$\longmapsto \rm \: \frac{3k - 6}{k + 1} = - 4 \\ \\ \longmapsto \rm \:3k - 6 = - 4k - 4 \\ \\ \longmapsto \rm \:7k = 2 \\ \\ \longmapsto \bf \:k = \frac{2}{7}$

2nd case:-

$\longmapsto \rm \: \frac{ - 8k + 10}{k + 1} = 6 \\ \\ \longmapsto \rm \: - 8k + 10 = 6k + 6 \\ \\ \longmapsto \rm \: -14k = - 4 \\ \\ \longmapsto \rm \: k = \frac{4}{14} \\ \\ \longmapsto \bf \: k = \frac{2}{7}$

So, k = 2/7 in both cases

So, the required ratio is 2:7

Hence, p divides AB in the ratio of 2:7

Answer 2

$\red{\underline{\underline{Answer:}}}$

$\sf{The \ line \ segment \ is \ divided \ in}$

$\sf{ratio \ of \ 2:7}$

$\sf\orange{Given:}$

$\sf{\implies{P(-4,6) \ divides \ the \ line \ joining}}$

$\sf{the \ points \ A(-6,10) \ and \ B(3-8)}$

$\sf\pink{To \ find:}$

$\sf{The \ ratio \ in \ which \ line \ is \ divided.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{Here, \ x=-4, \ y=6, \ x1=-6, \ y1=10, \ x2=3 \ and \ y2=-8}$

$\sf{By \ section \ formula}$

$\boxed{\sf{x=\frac{mx2+nx1}{m+n} \ ; \ y=\frac{my2+my1}{m+n}}}$

$\sf{\therefore{x=\frac{mx2+nx1}{m+n}}}$

$\sf{-4=\frac{m(3)+n(-6)}{m+n}}$

$\sf{-4(m+n)=3m-6n}$

$\sf{-4m-4n=3m-6n}$

$\sf{-4m-3m=-6n+4n}$

$\sf{-7m=-2n}$

$\sf{\frac{m}{n}=\frac{-2}{-7}}$

$\sf{\frac{m}{n}=\frac{2}{7}}$

$\sf{\therefore{m:n=2:7}}$

$\sf\purple{\tt{\therefore{The \ line \ segment \ is \ divided \ in}}}$

$\sf\purple{\tt{ratio \ of \ 2:7}}$