A point P ( x , y ) divides the line segment joining the points A ( 10 , - 2 ) and B ( 3 , 40 ) in the ratio " k : 1 " also P lies on the line " y - x - 9 = 0 " . Find the value of " k "
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Step-by-step explanation:
Given :-
point P ( x , y ) divides the line segment joining the points A ( 10 , - 2 ) and
B ( 3 , 40 ) in the ratio " k : 1 " also P lies on the line " y - x - 9 = 0 " .
To find :-
Find the value of " k " ?
Solution :-
Given points are A(10,-2) and B(3,40)
Let (x1, y1) = (10,-2) => x1 = 10 and y1 = -2
Let (x2, y2) = (3,40) => x2 = 3 and y2 = 40
Given ratio = k:1
Let m1:m2 = k:1 => m1 = k and m2 = 1
We know that
The section formula
The coordinates of the point P(x,y) =
((m1x2+m2x1)/(m1+m2), (m1y2+m2y1)/(m1+m2) )
On Substituting these values in the above formula then
=> ( {(k)(3)+(1)(10)}/(k+1), {(k)(40)+(1)(-2)}/(k+1) )
=> ({3k+10}/(k+1) , {40k-2}/(k+1))
We have P = ({3k+10}/(k+1) , {40k-2}/(k+1))
Given that P lies on y-x -9 = 0
If it lies on it ,P is a solution of the line
Then it must satisfies the given line
Put x = (3k+10)/(k+1) and
y = (40k-2)/(k+1) then
=> [(40k-2)/(k+1)]-[(3k+10)/(k+1)] -9 = 0
=> [(40k-2)-(3k+10)-9(k+1)]/(k+1) = 0
=> [(40k-2)-(3k+10)-9(k+1)] = 0
=> 40k-2-3k-10-9k-9 = 0
= > (40k-3k-9k)+(-2-10-9) = 0
=> (40k-12k) +(-21) = 0
=> 28k -21 = 0
=> 28k = 21
=> k = 21/28
=> k = 3/4
Therefore, k = 3/4
Answer:-
The value of k for the given problem is 3/4
Used formulae:-
Section formula:-
The coordinates of the point which divides the linesegment joining the points (x1, y1) and (x2, y2) in the ratio m1:m2 is
((m1x2+m2x1)/(m1+m2), (m1y2+m2y1)/(m1+m2) )