Question

If the line whose equation is 9x - 2ky + k = 0 passes through the point of intersection of lines whose equations are 2x
+ 7y + 22 = 0 and 5x - y - 19 = 0 then the value of k2 + k is equal to-​

Answer 1

Answer:

Lionel Andrés Messi[note 1] (Spanish pronunciation: [ljoˈnel anˈdɾez ˈmesi] (About this soundlisten);[A] born 24 June 1987) is an Argentine professional footballer who plays as a forward and captains both Spanish club Barcelona and the Argentina national team. Often considered the best player in the world and widely regarded as one of the greatest players of all time, Messi has won a record six Ballon d'Or awards,[note 2] and a record six European Golden Shoes. He has spent his entire professional career with Barcelona, where he has won a club-record 34 trophies, including ten La Liga titles

Step-by-step explanation:

Answer 2

SOLUTION

GIVEN

If the line whose equation is 9x - 2ky + k = 0 passes through the point of intersection of lines whose equations are 2x + 7y + 22 = 0 and 5x - y - 19 = 0

TO DETERMINE

The value of

$\sf{ {k}^{2} + k }$

EVALUATION

The equation of the given line is

$\sf{9x - 2ky + k = 0} \: \: ....(1)$

Which passes through the point of intersection of lines whose equations are

$\sf{2x + 7y + 22 = 0 } \: \: \: .....(2)$

$\sf{5x - y - 19 = 0 } \: \: \: .....(3)$

First we find the point of intersection of the lines given by Equation (2) & (3)

Multiplying Equation (2) by 5 and Equation (3) by (2) we get

$\sf{10x + 35y + 110 = 0 }$

$\sf{10x - 2y - 38= 0 }$

On substraction we get

$\sf{37y + 148 = 0}$

$\implies\sf{37y = - 148 }$

$\implies\sf{y = -4 }$

From Equation (3) we get

$\sf{5x = y + 19}$

$\implies\sf{5x = - 4 + 19 }$

$\implies\sf{5x = 15 }$

$\implies\sf{x = 3}$

So the point of intersection = ( 3, - 4 )

Now Equation (1) passes through the point ( 3, - 4)

So we get

$\sf{(9 \times 3) - 2k \times ( - 4) + k = 0}$

$\implies\sf{27 + 8k + k = 0}$

$\implies\sf{9 k = - 27}$

$\implies\sf{k = -3}$

Hence the required value

$\sf{ {k}^{2} + k}$

$= \sf{ {( - 3)}^{2} - 3}$

$= \sf{ 9 - 3}$

$= \sf{6}$

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