Question

OR
A point (6, a) divides the line joining A (-3,-1) and B (-8, 9) in the ratio k: 1. Find the value
of 'a' and 'k'.​

Answer 1

Answer:

a $\longmapsto$ -19

k $\longmapsto$ -9/14

Step-by-step explanation:

We've been given that there's a Line AB formed by the coordinates A(-3, -1) and B(-8, 9). This line has been divided in the ratio k : 1 by a point (6, a)

Let us name the point (6, a) as D.

We'll use the below formula to find out the value of 'k' and 'a'.

$\sf D(x, y) = \left(\dfrac{kx_2 + x_1}{k + 1} , \dfrac{ky_2 + y_1}{k + 1} \right)$

Now, substitute the coordinates of A, B & D in the above formula.

$\Longrightarrow \sf D(x, y) = \left(\dfrac{kx_2 + x_1}{k + 1} , \dfrac{ky_2 + y_1}{k + 1} \right)\\ \\ \\ \\\Longrightarrow \sf D(6, a) = \left(\dfrac{k(-8) + (-3)}{k + 1} , \dfrac{k(9) + (-1)}{k + 1} \right)\\ \\ \\ \\\Longrightarrow \sf D(6, a) = \left(\dfrac{-8k -3 \ }{ \ k + 1} , \dfrac{9k -1}{k + 1} \right)$

Here, the x coordinate (6) can be equated with (-8k - 3)/(k + 1) to find the value of 'k' and the y coordinate (a) can be equated with (9k - 1)/(k + 1) to find the value of 'a'.

Finding the value of 'k':

$\Longrightarrow \sf 6 = \dfrac{-8k - 3 \ \ }{k + 1}\\ \\ \\\Longrightarrow \sf 6(k + 1) = -8k - 3\\ \\ \\\Longrightarrow \sf 6k + 6 = -8k - 3\\ \\ \\\Longrightarrow \sf 6k + 8k = - 3 - 6\\ \\ \\\Longrightarrow \sf 14k = -9\\ \\ \\\Longrightarrow \sf k = -\dfrac{9}{14}$

Finding the value of 'a':

$\Longrightarrow \sf a = \dfrac{9k - 1}{k + 1}\\ \\ \\\Longrightarrow \sf a(k + 1) = (9k - 1)\\ \\ \\\Longrightarrow \sf a\left(\dfrac{-9}{14}\right) + a = 9\left(\dfrac{-9}{14}\right) - 1\\ \\ \\\Longrightarrow \sf -\dfrac{9a}{14}+ a = -\dfrac{81}{14} - 1\\ \\ \\\Longrightarrow \sf \dfrac{14a - 9a}{14} = \dfrac{-81 - 14}{14}\\ \\ \\\Longrightarrow \sf \dfrac{5a}{14} = \dfrac{-95}{14}\\ \\ \\\Longrightarrow \sf 5a = -95\\ \\ \\\Longrightarrow \sf a = \dfrac{-95}{5}\\ \\ \\\Longrightarrow \sf a = -19$

∴ k = -9/14

∴ a = -19.