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analytic geo

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A line passing through the intersection point of the two lines x+2y+3= 0 and 2x-y+a/2 = 0 also passes through two other points (9, −8)  and (−17, 9). What is the value of a?

Dec 26, 2020

#1
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x + 2y  + 3    = 0    (1)

2x  - y + a/2  = 0 ⇒    4x - 2y + a   =  0  (2)

5x + (3 +a)  = 0

x =  -(3 + a) /  5

Sub this into (1) for x

-(3 + a)   / 5  +  2y  + 3    =  0

2y  =  -3 + (3 + a)  / 5

y = (-12 + a)  / 10

So......the  intersection point  is   [x , y ]  =   [- (3 + a) /5 , (-12+ a) /10)]

The slope of the line through the given points is    [ 9 - - 8 ]  / [-17 - 9]  =  -17/26

And the equation of this line  is

y = (-17/26) ( x -9)  - 8

y = (-17/26) (x)  +  153/26   -208/26

y = (-17/26)x - 55/26

Note that this line  contains the intersection point of the  given lines....so   filling  in the values  for x and y

(-12 + a) / 10  = (-17/26) ( -(3 + a) / 5)  - 55/26

Multiply through by 260

26 (-12 + a)  = 2 ( -17) (-3-a) - 550

-312 + 26a  =  -34 (-3 - a) - 550

-312 +  26a  =  102 + 34a  - 550

-312 + 26a  =  34a -448

136  =  8a

a  =17

The intersection point  is   (-4, 1/2)

See the graph here  :   https://www.desmos.com/calculator/gsq2nbtdxq

Dec 26, 2020
edited by CPhill  Dec 26, 2020