Question

Solution find the envelope of the family of straight line y=mx+a/m

Answer 1

First of all,

An envelope of family of straight line is a curve that is tangent to a curve at some point

So,

If you look at the properties of parabola,

The equation of the tangents of the parabola is

y= mx + c

where c= a/m

Now,

if

c= a/m then only the line is tangent (1 intersection point)

If c>a/m then the line has no intersection points on parabola

If c<a/m then the line intersects the parabola twice

So,

the envelope for the straight line,

y=mx + a/m is

y² = 4ax