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

[karush]

https://www.physicsforums.com/attachments/2227
The region $R$, is bounded by the graphs of
$$ f(x)=x^2 -3$$ and $$g(x)=(x-3)^2$$,
and the line $$T$$, as is shown in the figure above.
$$T$$ is tangent to the graph of f at point $$(a,a^2-3) $$
and tangent to the graph of g at point $$(b,(b-3)^2)$$

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 $y=x-3.25$

 

[MarkFL]

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

 

[karush]

Re: tangent line to 2 parabola’s

not sure if I follow ...
but presume $T$ is both $2x(x-a)+(a^2-3)$ and $2x(x-b)+(b-3)^2$

 

[MarkFL]

Re: tangent line to 2 parabola’s

What I did was:

i) $$x^2-3=mx+b$$

standard form...

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

discriminant to zero...

$$m^2+4b+12=0$$

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

standard form...

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

discriminant to zero...

$$m^2+12m+4b=0$$

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

$$m^2+4b+12=m^2+12m+4b$$

Collect like terms:

$$12m=12$$

Divide through by 12:

$$m=1$$

And thus, we can find from either equation:

$$b=-\frac{13}{4}$$

Hence, the tangent line is:

$$y=x-\frac{13}{4}$$

 

[karush]

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