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Graphing parabolas

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The graph of the line y = 3x + a intersects the graph of the parabola y = x^2 + x in two points. If the distance between these points is sqrt(30), what is the value of a? Express your answer as a common fraction.

Jul 13, 2022

#1
+125627
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Set the equations  =   to find the x coordinates (in terms of a) of the intersections of these two functions

3x + a =   x^2 +  x          rearrange as

x^2 - 2x   =   a              complete the  square on x

x^2  -2x + 1 =  a + 1

(x  - 1)^2  = a + 1           take both roots

x -  1 =  ± √ (a + 1)

x = ± √ (a + 1)  + 1

Plugging these two x values into  the  equation of the  line we  have that

y =  3 ( sqrt (a + 1) + 1) + a        or      y =  3 ( -sqrt (a + 1) + 1) + a

y =  3sqrt (a + 1) + a + 3                      y =  -3sqrt (a + 1) + a + 3

Using  the square of the distance formula we have that

( (sqrt (a + 1) + 1)  - ( -sqrt (a + 1) + 1) )^2  +  ( (3sqrt (a + 1) + a + 3) - (-3sqrt (a + 1) + a + 3))^2 = 30

Simplify

( 2 sqrt ( a + 1) )^2   +  ( 6 sqrt (a + 1 )^2     =  30

4 (a + 1)  + 36 (a + 1)  = 30

40a  + 40  =  30

40a =  -10

a =  -1/4

Here's a graph :

Points  A and B  are the intersections of  the functions.......the midpoint of the segment connecting A and B  = (1, 2.75)

A circle with this center and a diameter of sqrt (30) = radius of sqrt (30) /2   will pass through both points

Jul 15, 2022
edited by CPhill  Jul 15, 2022
#2
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Nice work Chris,

Its a pity that these question askers are mostly just figments of the internets imagination.

Jul 16, 2022