Question

The midpoint P of the line segment joining the points A(–10, 4) and B(–2, 0) lies on the line segment joining the points C(–9, –4) and D(–4, y). Find the ratio in which P divides CD. Also, find the value of y.

Answer 1

The midpoint of AB is `((-10-2)/2 , (4+10)/2) = P(-6,2).`

Let k be the ratio in which P divides CD. So

`(-6,2) = ((k(-4)-9)/(k+1) , (k(y)-4)/(k+1))`

`⇒ (k(-4)-9)/(k+1) = -6 and (k(y)-4)/(k+1) = 2`

`⇒ k = 3/2`

Now, substituting `k= 3/2 " in" (k(y)-4)/(k+1) = 2, ` we get

`(y xx3/2-4)/(3/2+1) = 2 `

`⇒  (3y-8)/5 =2`

`⇒ y = (10+8)/3 = 6`

Hence, the required ratio is 3:2and y = 6

Answer 2

It is given that P is the mid-point of the line segment joining the points A(−10, 4) and B(−2, 0).
∴ Coordinates of P = \[\left( \frac{- 10 + \left( - 2 \right)}{2}, \frac{4 + 0}{2} \right) = \left( \frac{- 12}{2}, \frac{4}{2} \right) = \left( - 6, 2 \right)\]

Suppose P divides the line segment joining the points C(−9, −4) and D(−4, y) in the ratio k : 1.
Using section formula, we get
Coordinates of P = \[\left( \frac{- 4k - 9}{k + 1}, \frac{ky - 4}{k + 1} \right)\]

\[\therefore \left( \frac{- 4k - 9}{k + 1}, \frac{ky - 4}{k + 1} \right) = \left( - 6, 2 \right)\]

\[ \Rightarrow \frac{- 4k - 9}{k + 1} = - 6 \text{ and }  \frac{ky - 4}{k + 1} = 2\]

Now,

\[\frac{- 4k - 9}{k + 1} = - 6\]
\[ \Rightarrow - 4k - 9 = - 6k - 6\]
\[ \Rightarrow 2k = 3\]
\[ \Rightarrow k = \frac{3}{2}\]

So, P divides the line segment CD in the ratio 3 : 2.
Putting k = \[\frac{3}{2}\]  in

\[\frac{ky - 4}{k + 1} = 2\] , we get

 

\[\frac{\frac{3y}{2} - 4}{\frac{3}{2} + 1} = 2\]

\[ \Rightarrow \frac{3y - 8}{5} = 2\]

\[ \Rightarrow 3y - 8 = 10\]

\[ \Rightarrow 3y = 18\]

\[ \Rightarrow y = 6\]

Hence, the value of y is 6.