Question

Show that the equation of the parabola in standard from is $y^2=4ax$.

Answer 1

Solution:

Let $S$ be the focus,ZM the directrix and P the moving point.

Draw $SZ$ perpendicular from $S$ on the directrix. Then $SZ$ is the axis of the parabola. Now the middle point of $SZ$,say $A$, will lie on the locus of $P$, i.e., $AS=AZ$. Take A as the origin, the x-axis along $AS$, and the y-axis along the perpendicular to $AS$ at $A$, as in the figure.

Let $AS=a$, so that $ZA$ also $a$, Let $(x,y)$ be the coordinates of the moving point $P$. Then $MP=ZN=ZA+AN=a+x$. But by definition $MP=PS⟹M{P}^2=P{S}^2$

so that, $(a+x{)}^2=(x-a{)}^2+y^2.$

Hence, the standard equation of the parabola is $y^2=4ax$.