How Do You Find a Common Tangent Line to Two Parabolas?

In summary, the line T is tangent to the graphs of f and g at points (a,a^2-3) and (b,(b-3)^2), respectively.
  • #1
karush
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The region $R$, is bounded by the graphs of
\(\displaystyle f(x)=x^2 -3\) and \(\displaystyle g(x)=(x-3)^2\),
and the line \(\displaystyle T\), as is shown in the figure above.
\(\displaystyle T\) is tangent to the graph of f at point \(\displaystyle (a,a^2-3) \)
and tangent to the graph of g at point \(\displaystyle (b,(b-3)^2)\)

a. Show that $a=b-3$
b. Show the numerical value of $a$ and $b$
c. Write the equation of $T$
d. Setup but do not calculate the integral for the region $R$.

this is a common problem, and I looked at several solutions via several forums but only got lost on this. I know that the slope of the $2$ parabola's is $2x$ and since it is a line the slope is the same. I did try using $y=mx+b$ but with $3$ variables $a,b$ and $x$ it was hard work with. If I could see how the equation of $T$ is derived then the other questions I should be able to answer.
the answer is not given but the line eq appears to be close to $y=x-3.25$:cool:
 
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  • #2
Re: tangent line to 2 parabola’s

What I would for part c) is let the tangent line be $y=mx+b$. Next, equate this line to the two parabolas in turn and require the resulting discriminants to both be zero, giving two equations in two unknowns from which $m$ and $b$ can be easily found. Once you have this, then the other parts of the question should be a piece of cake. :D
 
  • #3
Re: tangent line to 2 parabola’s

not sure if I follow ...
but presume $T$ is both $2x(x-a)+(a^2-3)$ and $2x(x-b)+(b-3)^2$
 
  • #4
Re: tangent line to 2 parabola’s

What I did was:

i) \(\displaystyle x^2-3=mx+b\)

standard form...

\(\displaystyle x^2-mx-(b+3)=0\)

discriminant to zero...

\(\displaystyle m^2+4b+12=0\)

ii) \(\displaystyle (x-3)^2=mx+b\)

standard form...

\(\displaystyle x^2-(m+6)x+(9-b)=0\)

discriminant to zero...

\(\displaystyle m^2+12m+4b=0\)

Now, since both discriminants are zero, we may equate them and write:

\(\displaystyle m^2+4b+12=m^2+12m+4b\)

Collect like terms:

\(\displaystyle 12m=12\)

Divide through by 12:

\(\displaystyle m=1\)

And thus, we can find from either equation:

\(\displaystyle b=-\frac{13}{4}\)

Hence, the tangent line is:

\(\displaystyle y=x-\frac{13}{4}\)
 
  • #5
Re: tangent line to 2 parabola’s

OK, I see now, will try to get the rest of it. that was a big help
BTW I have an incredible view of Pearl Harbor here at LCC:cool:
 
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FAQ: How Do You Find a Common Tangent Line to Two Parabolas?

What is a tangent line?

A tangent line is a straight line that touches a curve at one point, without crossing through the curve. It represents the instantaneous slope of the curve at that point.

What are parabolas?

Parabolas are a type of curve that appear in the shape of a "U" or an "upside-down U". They are defined by a quadratic function, which takes the form y = ax^2 + bx + c.

How do you find the tangent line to a parabola?

To find the tangent line to a parabola, you must first find the point of tangency, which is the point where the tangent line touches the curve. Then, you can use the derivative of the parabola to find the slope of the tangent line at that point. Finally, you can use the point-slope formula to write the equation of the tangent line.

Can a tangent line intersect two parabolas?

Yes, it is possible for a tangent line to intersect two parabolas. This can occur when the two parabolas are tangent to each other at a point, and the tangent line passes through that point. In this case, the tangent line is tangent to both parabolas at that point.

What is the significance of the tangent line to 2 parabolas?

The tangent line to 2 parabolas is significant because it represents the instantaneous rate of change of both parabolas at the point of tangency. It can also be used to approximate the behavior of the parabolas near that point.

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