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The line x = k intersects the graph of the parabola x = -y^2 - 3y - 5 at exactly one point. What is k?

 Jun 15, 2022
 #1
avatar+13728 
+3

 What is k?

 

Hello Guest!

 

\( x = -y^2 - 3y - 5\\ \frac{dx}{dy}=-2y-3=0\\ y_{max}=-\frac{3}{2}\\ x_{max}=k=-(-\frac{3}{2})^2-3\cdot(-\frac{3}{2})-5\\ \color{blue}k=-2.75\)

laugh  !

 Jun 15, 2022
 #2
avatar+123309 
+1

x = -y^2  - 3y    -5

 

This is a parabola that opens to the left

 

The only way a line can intersect the parbola at one point is if it interesects it at the vertex

 

The y coodinate of the vertex is     - (-3) / (2 * -1)  =  3/2

 

So  the x coordinate of the vertex = k    is

 

-(3/2)^2 - 3 (3/2)  - 5

 

-9/4 + 9/2   - 20/4   =

 

9/4  - 20/4  = 

 

-11/4   =  -2.75  =   k

 

 

 

cool cool cool

 Jun 15, 2022

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