+0  
 
0
121
2
avatar

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
avatar+124676 
+2

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

 

 

cool cool cool

 Jul 15, 2022
edited by CPhill  Jul 15, 2022
 #2
avatar+118139 
0

Nice work Chris,

 

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

 Jul 16, 2022

16 Online Users

avatar