What is the common tangent to the parabolas #y^2=4ax# and #x^2=4by# ?

Answer 1

#root(3)by+root(3)ax+root(3)(a^2b^2)=0#

Let the common tangent be #y=mx+c#
Let us find the points of intersection of this with parabola #y^2=4ax#. These are given by
#(mx+c)^2=4ax# or #m^2x^2+x(2mc-4a)+c^2=0#

and as it is a tangent, we should have discriminant zero i.e.

#(2mc-4a)^2-4m^2c^2=0# or #16a^2-16amc=0# or #a=mc# (A)
Similarly tangency of #y=mx+c# with #x^2=4by# leads to points of intersection given by
#x^2-4bmx-4bc=0# and as it is tangent we should have
#16b^2m^2+16bc=0# i.e. #bm^2+c=0# (B)
Putting value of #m# from (A) in (B), we get #(ba^2)/c^2+c=0# or #c^3=-ba^2# i.e. #c=-root(3)(ba^2)#
and putting this back in (A) we get #m=-a/root(3)(ba^2)=-root(3)(a/b)#
and hence equation of tangent is #y+root(3)(a/b)x+root(3)(ba^2)=0# or #root(3)by+root(3)ax+root(3)(a^2b^2)=0#
Below we show the graph of parabolas and the tangent, when #a=1# and #b=2#

graph{(y^2-4x)(x^2-8y)(root(3)2y+x+root(3)4)=0 [-5.416, 4.584, -2.8, 2.2]}

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7