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1.For what value of $c$ will the circle with equation $x^2 - 10x + y^2 + 6y + c = 0$ have a radius of length 1?

2.Two parabolas are the graphs of the equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons.

Guest Feb 21, 2018
 #1
avatar+7154 
+2

1.

x2 - 10x + y2 + 6y + c   =   0    Let's get this equation into standard form.

                                                Subtract  c  from both sides of the equation.

x2 - 10x + y2 + 6y   =   0 - c     Add  25  and  9  to both sides to complete the squares on the left.

 

x2 - 10x + 25 + y2 + 6y + 9   =   0 - c + 25 + 9      Factor each perfect square trinomial on the left.

 

(x - 5)2  +  (y + 3)2   =   0 - c + 25 + 9

 

(x - 5)2  +  (y + 3)2   =   -c + 34

 

Now that the equation is in this form, we can see that

 

(the radius)2  =  -c + 34        If the radius is  1 ,  then

 

12   =   -c + 34

 

1   =   -c + 34

 

c  =  33

hectictar  Feb 21, 2018
 #2
avatar+7154 
+2

2.

We want to find the solutions to this system of equations:

y   =   2x2 - 7x + 1

y   =   8x2 + 5x + 1

 

8x2 + 5x + 1   =   2x2 - 7x + 1

                                                    Subtract  2x2  from both sides.

6x2 + 5x + 1   =   -7x + 1

                                                    Add  7x  to both sides.

6x2 + 12x + 1   =   1

                                                    Subtract  1  from both sides.

6x2 + 12x   =   0

                                                    Factor  6x  out of both terms on the left side.

6x(x + 2)   =   0

                                                    Set each factor equal to zero.

6x  =  0     or     x + 2  =  0

x   =   0     or     x   =   -2

 

Using one of the equations, find  y  when  x = 0 .

 

When  x = 0 ,   y   =   2(0)2 - 7(0) + 1   =   1

 

Find  y  when  x = -2 .

 

When  x = -2 ,   y   =   2(-2)2 - 7(-2) + 1   =   8 + 14 + 1   =   23

 

The points of intersection are   (0, 1)   and   (-2, 23)

hectictar  Feb 21, 2018
edited by hectictar  Feb 21, 2018

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