find the envelope of y=mx+1/m where m is the parameter
Answers
Answer:
Envelope of one-parameter family of curve is the locus of the limiting positions of the points of intersection of any two members of the family when one of the family tends to coincide with the other family which is kept fixed. It is done by Eliminating 'mm' which is assumed to be parameter here from given and partial differentiation equations.
We have, y=mx+\frac{a}{m}y=mx+
m
a
(*)
Partial differentiate with respect to mm: 0=x+a(-\frac{1}{m^2})0=x+a(−
m
2
1
). It follows that m=\sqrt{\frac{a}{x}}m=
x
a
or m=-\sqrt{\frac{a}{x}}m=−
x
a
.
Substituting in equation (*) , y=\sqrt{ax}+\sqrt{ax}=2\sqrt{ax}y=
ax
+
ax
=2
ax
. When we take negative value of mm, then y=-2\sqrt{ax}y=−2
ax
. Combining these two, we get y^2=4ax.y
2
=4ax.
This represents Parabola which is required envelope of family of straight lines y=mx+\frac{a}{m}y=mx+
m
a
.