Question

determine the ratio in which the point p(a, 2) divides the join of A(-4, 3) and B(2, -4) also the value of a​

Answer 1

Answer

The ratio in which P divides the line joining A and B is 1:6

And the value of a is -22/7

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

• The given points are A(-4 , 3) and (2 , -4)
• The point P(a , 2) divides the line joining the points A and B

$\large \bf \underline{To \: Find : }$

• The ratio in which P divides the line joining the points A and B
• The value of a

$\bf \large \underline{Formula \: to \: be \: used : }$

$\sf If \: (x_{1} , y_{1} )\: and \: ( x_{2} , y_{2} ) \: are \ two \ points \ \\ \sf and \ (x , y) \ divides \ these \ points \ in \\ \sf ratio \ m:n \: then :$

$\tt x = \dfrac{mx_{2} + nx_{1}}{m+n } \: \: , \: y = \dfrac{my_{2} + ny_{1}}{m+n}$

$\bf \large \underline{Solution :}$

Let us consider that the point P(a , 2) divides the line joining A(-4 , 3) and B(2 , -4) in the ratio m:n

Thus we have from section formula ,

$\sf\rightarrow a = \dfrac{2m -4 n}{m+n } \: \: , 2 = \dfrac{-4m +3n}{m + n }$

Now taking ,

$\sf\implies 2 = \dfrac{-4m +3n}{m+n} \\\\ \sf\implies 2(m+n) = -4m +3n \\\\ \sf\implies 2m + 2n = -4m + 3n \\\\ \sf\implies 6m = n \\\\ \sf\implies \dfrac{m}{n} = \dfrac{1}{6} \\\\ \sf\implies m:n = 1:6$

Therefore , required ratio is 1:6

We are given to find the value of a , so :

$\sf \implies a = \dfrac{2\times 1 + (-4)\times 6}{1+6} \\\\ \sf\implies a = \dfrac{2 - 24}{7} \\\\ \sf\implies a = \dfrac{-22}{7}$