Question

Find ratio in which the point (-3,k) divides the line segment joining points (-5,-4) and (-2,-3). also find value of k .

Answer 1

Let's get started...

Let, the point (-5,-4) be named as A and point (-2,-3) be B and the point which divides the line segment AB be P(-3,k). The point P divides the line segment AB in the ratio in the m:n.
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The segment will look like :-

A._______m_______P_______n______.B

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Given data,
x1= -5
x2= -2
y1= -4
y2= -3

Ratio of division of line= m:n
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Formula for finding coordinates of a point which divides a line in the ratio m:n ===>

$P( - 3,k) = ( \frac{nx1 + mx2}{m + n} , \: \frac{ny1 + my2}{m + n} )$
$P( - 3,k) = (\frac{( - 5 \times n) + (- 2 \times m)}{m + n} , \: \frac{ (- 4 \times n) + ( - 3 \times m)}{m + n} )$
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On comparing the above,

$- 3 = \frac{ - 5n - 2m}{m + n}$
$- 3m -3n = - 5n - 2m$
$5n - 3n = - 2m + 3m \\ 2n = m \\ \frac{m}{n} = \frac{2}{1} \\ m:n \: = 2:1$
So, the ratio in which the point (-3,k) divides the line segment joining points (-5,-4) and (-2,-3) is m:n ===> 2 : 1

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Now, to find the value of k, compare k with the ordinate in the following equation...
$P( - 3,k) = (\frac{( - 5 \times n) + (- 2 \times m)}{m + n} , \: \frac{ (- 4 \times n) + ( - 3 \times m)}{m + n} )$
$k = \frac{ - 4n - 3m}{m + n}$
Put the value of m and n which is calculated above in the solution...

$k = \frac{ - 4 - 6}{3}$

$k = \frac{ - 10}{3}$
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So, the value of k is
$k = \frac{ - 10}{3}$
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Hence, at last we got the ratio in which the point (-3,k) divides the line segment joining points (-5,-4) and (-2,-3) which is 2 : 1.

And, value of k is -10/3

MESSAGE ME IF ANY DOUBT....
THANKS...

Answer 2

Let (-3,k) divide the line segment joining the points (-5,-4)and (-2,3) in the ratio m₁:m₂.Using the section formula , we get

(-3,k)=[(-2m₁-5m₂)/(m₁+m₂),(3m₁-4m₂)/(m₁+m₂)]

-3= (-2m₁-5m₂)/(m₁+m₂)

-3m₁-3m₂ =-2m₁-5m₂

m₁=2m₂ ,m₁/m₂=2:1

k = (3m₁-4m₂)/(m₁+m₂)

k =(6m₂-4m₂)/3m₂= 2m₂/3m₂=2/3