Question

Find the ratio in which the line 2x + y = 4
divides the line segment joining the points P
(2,-2) and Q (3, 7).​

Answer 1

Answer:

Ratio is 9:2

Step-by-step explanation:

Interesting Question

line PQ :

y + 2 = (9/1)(x - 2)

y + 2 = 9x - 18

y = 9x - 20

Given line :

y = -2x + 4

Let's calculate their point of intersection.

9x - 20 = -2x + 4

11x = 24

x = 24/11

y = -48/11 + 4

y = -4/11

So, the given line interects line segment PQ in

(24/11, -4/11)

Let it divide PQ in the ratio m : 1

Then,

(m×2 + 1×3)/(m+1) = 24/11 ____(1)

and

(m×(-2) + 1×7)/(m+1) = -4/11 _____(2)

eqn (1) ÷ eqn (2)

(2m + 3)/(-2m + 7) = -6

2m + 3 = 12m - 42

10m = 45

m = 9/2

So, it divides PQ in the ratio 9/2:1 or 9:2

Answer 2

Answer:

given : P (2, -2) , Q (3, 7)

equation of line 2x + y = 4

let the line divide it in the ratio k:1 at point A

then , m = k and n = 1

A x = [(m x₂ + n x₁) / m + n]

= (k*3 + 1*2)/(k+1)

=> (3k + 2)/(k+1)

A y = [k*7 + 1*(-2)]/ (k+1)

=> (7k - 2)/(k+1)

coordinates of A [ (3k+2)/(k+1) , (7k-2)/(k+1) ]

coordinates of A must satisfy the equation

2x + y = 4

substituting these values, we get

2 [ (3k+2)/(k+1) + (7k-2)/(k+1) = 4

=> 2 [ 3k+2+7k-2 ]/ (k+1) = 4/1

=> (6k+14k)/(k+1) = 4/1

=> 20k = 4k + 4

=> 20k - 4k = 4

=> 16k = 4

=> 4k = 1

=> k = 1/4

therefore ratio is k:1 => 1/4:1

=> 1 : 4 Answer