Question

In what ratio does the point (- 4,6) divide the line segment joining the
points A(-6, 10) and B(3,- 8)? *​

Answer 1

Given ,

• The point P(- 4,6) divide the line segment joining the points A(-6, 10) and B(3,- 8)

Let ,

The point P divide the line segment AB in the ratio k : 1

We know that , the section formula is given by

$\boxed{ \sf{x = \frac{m x_{2} +n x_{1} }{m + n} \: , \: y = \frac{m y_{2} +n y_{1} }{m + n}}}$

Thus ,

$\sf \mapsto -4 = \frac{ -6k + 3}{k + 1} \\ \\ \sf \mapsto</p><p>-4k - 4 = -6k + 3 \\ \\ \sf \mapsto</p><p>2k = 7 \\ \\ \sf \mapsto</p><p>k = \frac{7}{2} </p><p></p><p></p><p></p><p></p><p>$

Therefore ,

• The point P divide the line segment AB in the ratio 7 : 2

Answer 2

GIVEN :-

• segment joining the points are :- (-6, 10) and B(3,-8) is divided by (-4, 6)

TO FIND :-

• The ratio in which the line segment joining

SOLUTION :-

let the ratio = m : n

hence ,

$\implies \rm{ \bold {a \: ( - 6 ,1)≡(x _{1}, \:y _{1})}}$

$\implies \rm{ \bold {b\: ( 3 , - 8)≡(x _{2}, \:y _{2})}}$

$\implies \rm{ \bold {c\: ( - 4 \: , 6)≡(x _{3}, \:y _{3})}}$

now we know the section formula that ,

$\implies \boxed{\rm { \dfrac{mx _{2} + \: nx _{1}}{m + n}, \dfrac{my_{2} + \: ny _{1}}{m + n} = (x _{3}, \:y _{3})}}$

$\implies \rm { \dfrac{mx _{2} + \: nx _{1}}{m + n}, \dfrac{my_{2} + \: ny _{1}}{m + n} = ( - 4 \: , \:6)}$

we can solve by both by ‘ X ’ or ‘ y ’ so we are take X

$\implies \rm { \dfrac{mx _{2} + \: nx _{1}}{m + n}= - 4 \: }$

$\implies \rm { mx _{2} + \: nx _{1} = - 4 \: (m + n)\: }$

now put the value of X2 and X1 in the following

$\implies \rm { m(3) + \: n( - 6) = - 4 \: (m + n)\: }$

$\implies \rm { 3m - 6n = - 4 m - 4 n\: }$

$\implies \rm { 3m + 4m = 6n - 4 n\: }$

$\implies \rm { 7m = 2 n\: }$

$\implies \rm { \dfrac{m}{n} = \: \dfrac{2}{7} }$

$\implies \boxed{ \boxed { \rm { m : n = 2 : 7 }}}$

OTHER INFORMATION :-

Section Formula :

• When a point C divides a segment AB in the ratio m:n, we use the section formula to find the coordinates of that point. The section formula has 2 types. These types depend on the position of point C. It can be present between the 2 points or outside the segment.

The two types are:

• Internal Section Formula

• External Section Formula

• Internal Section Formula

• Also known as the Section Formula for Internal Division. When the line segment is divided internally in the ration m:n, we use this formula. That is when the point C lies somewhere between the points A and B.