I am trying to explore whether it is possible to find a common tangent to two parabolas? And i can't work out if it is always possible.
I started with these two:
y= (x-2)^2 +3
and
y= -(x+2)^2 -3
By looking for a value at which the gradient of a tangent will be the same, and some messing with algebra, I established the line y= (sqrt28- 4)x is a common tangent to both parabolas.
Now, i can't work out if it is only possible to find common tangents to parabolas only if one 'upside down', like my example above?
My algebraic method fails if they are not, but visually I feel it must be possible?
( By common tangent i mean if one parabola has a tangent line at a particular point , the same line is also a tangent to the other parabola at some point on the curve..hope that makes sense)
Any thoughts?
I started with these two:
y= (x-2)^2 +3
and
y= -(x+2)^2 -3
By looking for a value at which the gradient of a tangent will be the same, and some messing with algebra, I established the line y= (sqrt28- 4)x is a common tangent to both parabolas.
Now, i can't work out if it is only possible to find common tangents to parabolas only if one 'upside down', like my example above?
My algebraic method fails if they are not, but visually I feel it must be possible?
( By common tangent i mean if one parabola has a tangent line at a particular point , the same line is also a tangent to the other parabola at some point on the curve..hope that makes sense)
Any thoughts?