Help with this quadratics question please! :) (1 Viewer)

[hayhay2012]

A parabola has its equation in the form y=Ax^2 where A is a constant. The line y=10x +10 is a tangent to the parabola.
1)Find the value of A
2) Find the coordinates of the focus and the equation of the directrix of the parabola

 

[Sy123]

1)
Here is a neat way to do it rather than differentiating and stuff
We know that since y=10x+10 is a tangent, it must have only one solution to the equation:



That means the discriminant of this equation will be 0.



Find the discriminant



Equate to 0 since it has one solution



2)

Its in the form x^2=4ay, so our a must be:



Vertex is at (0,0), its an upside down parabola, hence our focus is at:

Focus(0, -1/10)

Directrix: y=1/10

 

[hayhay2012]

thanks heaps! thats cool, how would you do the differentiating way though? (im only a 2u so im worried i dont know that many cool tricks)

1)
Here is a neat way to do it rather than differentiating and stuff
We know that since y=10x+10 is a tangent, it must have only one solution to the equation:



That means the discriminant of this equation will be 0.



Find the discriminant



Equate to 0 since it has one solution



2)

Its in the form x^2=4ay, so our a must be:



Vertex is at (0,0), its an upside down parabola, hence our focus is at:

Focus(0, -1/10)

Directrix: y=1/10

 

[Sy123]

thanks heaps! thats cool, how would you do the differentiating way though? (im only a 2u so im worried i dont know that many cool tricks)

I originally thought of differentiating equating the gradient to 10, then finding the point of intersection of of the tangent and the parabola but it leads you nowhere really. My way is 2U and probably want they wanted you to do

 

[hayhay2012]

I originally thought of differentiating equating the gradient to 10, then finding the point of intersection of of the tangent and the parabola but it leads you nowhere really. My way is 2U and probably want they wanted you to do

OK I get it now, since a tangent only hits a parabola once it has only one root/solution
thanks