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In what ratio does the point P(−4, y) divides the line segment joining the points A(−6, 10) and B(3, −8)? Hence find the value of y.

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#### Solution

Let P divides the line segment AB in the ratio k : 1

Using section formula

x = `(m_1x_2 + m_2x_1)/(m_1+m_2), y = (m_1y_2 + m_2y_1)/(m_1+m_2)`

A(-6, 10) and B(3, 8)

m_{1} : m_{2} = k : 1

plugging values in the formula we get

- 4 = `( k xx 3 + 1 xx (-6))/(k + 1), y = ( k xx (- 8) + 1 xx 10)/(k + 1)`

- 4 = `( 3k - 6)/(k + 1), y = (-8k + 10)/(k + 1)`

Considering only x coordinate to find the value of k

- 4k - 4 = 3k - 6

- 7k = - 2

k = `2/7`

k : 1 = 2 : 7

Now, we have to find the value of y

so, we will use section formula only in y coordinate to find the value of y.

y = `(2 xx (- 8) + 7 xx 10)/(2 + 7)`

y = `( - 16 + 70 )/(9)`

y = 6

Therefore, P divides the line segment AB in 2 : 7 ratio

And value of y is 6.

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