# The Midpoint P of the Line Segment Joining Points A(-10, 4) and B(-2, 0) Lies on Line Segment Joining the Points C(-9, -4) and D(-4, Y). Find Ratio in Which P Divides Cd. Also, Find the Value of Y. - Mathematics

The midpoint P of the line segment joining 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.

#### Solution 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

#### Solution 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.

Concept: Coordinate Geometry
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Chapter 6: Co-Ordinate Geometry - Exercise 6.3 [Page 31]

#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.3 | Q 57 | Page 31
RS Aggarwal Secondary School Class 10 Maths
Chapter 16 Coordinate Geomentry
Exercises 2 | Q 34

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