Question

In what ratio does the point p(1, 2)
divide the join of A[-2, 8) and
B(-6,-4)​

Answer 1

Point P(1,2) divide the line joining points A(-2,8) and B(-6,-4).

Ratio in which point P divide the line segment AB.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

☯ Using section formula,

$\star\;{\boxed{\sf{\purple{(x,y) = \bigg( \dfrac{m_1x_2 + m_2x_1}{m_1 + m_2}\;,\; \dfrac{m_1y_2 + m_2y_1}{m_1 + m_2} \bigg)}}}}$

where, (x,y) give the coordinates of point which divide the line segment joining points $\sf (x_1 , y_1)$ and $\sf (x_2 , y_2)$ in ratio $\sf m_1 : m_2.$

☯ Let the ratio in which point be divide the line segment AB be k : 1.

★ Now, Putting values in formula,

$:\implies\sf (1,2) = \bigg( \dfrac{(k)(-6) + (1)(-2)}{(k + 1)}\;,\; \dfrac{(k)(-4) + (1)(8)}{k + 1} \bigg)$

$\qquad \qquad \::\implies\sf (1,2) = \bigg( \dfrac{- 6k - 2}{k + 1}\;,\; \dfrac{- 4k + 8}{k + 1} \bigg)$

Therefore,

Comparing LHS and RHS

$\qquad:\implies\sf 1 = \bigg( \dfrac{- 6k - 2}{k + 1} \bigg)\;,\;2 = \bigg( \dfrac{- 4k + 8}{k + 1} \bigg)$

$\qquad\qquad \qquad:\implies\sf 2 = \bigg( \dfrac{- 4k + 8}{k + 1} \bigg)$

$\qquad\qquad \: \quad:\implies\sf 2(k + 1) = - 4k + 8$

$\qquad\qquad\qquad:\implies\sf 2k + 2 = - 4k + 8$

$\qquad\qquad\quad \:\:\quad:\implies\sf 2k + 4k = 8 - 2$

$\qquad\qquad\qquad\qquad\quad:\implies\sf 6k = 6$

$\qquad\qquad\qquad\qquad\quad:\implies\sf k = \cancel{ \dfrac{6}{6}}$

$\qquad\qquad\qquad\qquad\quad:\implies{\boxed{\frak{\pink{k = 1}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;AB\;line\; segment\;is\;divided\;by\;point\;P\;in\;ratio,\;k : 1 = \bf{1 : 1}.}}}$

Answer 2

Answer:

Step-by-step explanation:Follow me