Question

Let P be the point of intersection of the lines 2x + 3y = 3a and 3x – 4y = 8 and Q be the point of intersection of the lines x + 2y = 5 and 3x – y = 4. If the line passing through P and Q has slope 2, then the value of a is

Answer 1

Given :  P  point of intersection of the lines 2x + 3y = 3a and 3x – 4y = 8 and Q  point of intersection of the lines x + 2y = 5 and 3x – y = 4

the line passing through P and Q has slope 2

To Find : value of a

Solution:

2x + 3y = 3a     ..Eq1    

3x – 4y = 8        ..Eq2

4 * Eq1 + 3 * eq2

 8x + 9x  = 12a  + 24

=> 17x = 12(a + 2)

=> x = 12(a + 2)/17

3* eq1  - 2 * Eq2

=> 17y  = 9a - 16

=> y = (9a - 16)/17

P (  12(a + 2)/17 , (9a - 16)/17)

x + 2y = 5

3x – y = 4

=> x = 13/7  , y = 11/7

Q = ( 13/7 , 11/7)

Slope of PQ  =   ( (9a - 16)/17 - 11/7) / ( 12(a + 2)/17 -  13/7)  =  2

=>  (63a - 112 - 187) = 2 ( 84a + 168 - 221 )

=> 63a - 299 = 168a - 442

=> 105a = 143

=> a = 143/105

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