Question

In what ratio does the point $p(2 -5)$ divide the line segment joining $a(-3 5)$ and $b(4 -9)$.
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Answer 1

Answer:

5:2

Step-by-step explanation:

let point p divide the line segment internally by a(x1 y1) and b(x2 y2) in k:1 ratio

then coordinate of point p is given by

( (kx2+x1)/(k+1) , (ky2+y1)/(k+1) )

as per question

for x co-ordinate

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

4k-3=2k+2

2k=5

k=5/2

solving for y co-ordinate

(-9k+5)/(k+1) = -5

-9k+5 = -5k -5

4k=10

k=5/2

ratio is k:1

i.e 5/2 : 1

or, 5:2

it divides the line segment in ratio 5:2

Answer 2

Solution :-

We have got the points of the segment

♦ a ( -3 , 5 )

♦ b ( 4 , -9 )

The Section point

♦ p ( 2 , -5 )

First let us suppose that the point

p( 2 , -5) divides the line segment joining

a ( -3 , 5 ) and b ( 4 , -9 ) in the ratio of

k:1 .

Now as the section formula ( for internal division ) in the ratio of m : n .

$= \left( \dfrac{ mx_2 + nx_1 }{m + n} \: , \: \dfrac{ my_2 + ny_1 }{m + n} \right)$

So our x coordinate

$\rightarrow \dfrac{ k(4) + (1)(-3)}{k + 1} = 2$

$\rightarrow \dfrac{ 4k - 3 }{k + 1} = 2$

$\rightarrow 4k - 3 = 2k + 2$

$\rightarrow 2k = 5$

$\rightarrow k = \dfrac{5}{2}$

Now our y coordinate

$\rightarrow \dfrac{ k(-9) + (1)(5)}{k + 1} = -5$

$\rightarrow \dfrac{ 5 - 9k }{k + 1} = -5$

$\rightarrow 5 - 9k = -5k - 5$

$\rightarrow 4k = 10$

$\rightarrow k = \dfrac{5}{2}$

Now the ratio :-

= k : 1

And as

$\rightarrow k = \dfrac{5}{2}$

Therefore

$\rightarrow k : 1 = \dfrac{5}{2} : 1$

Therefore ratio

$\huge{\boxed{\it{= 5 : 2}}}$