Find minimum distance between parabolas x^2 and -(x - 6)^2

[fatlord]

Given: y = x^(2) and y = -(x-6)^(2)

y = x^(2) points upward and y = -(x-6)^(2) points downward.

Find the minimum distance between them.

can someone give me any hints on how to solve this.

 

[Deleted member 4993]

Given: y = x^(2) and y = -(x-6)^(2)

y = x^(2) points upward and y = -(x-6)^(2) points downward.

Find the minimum distance between them.

can someone give me any hints on how to solve this.

choose point A (x[sub:24n48gzm]1[/sub:24n48gzm],x[sub:24n48gzm]1[/sub:24n48gzm][sup:24n48gzm]2[/sup:24n48gzm]) on parabola (1) and choose a point B (x[sub:24n48gzm]2[/sub:24n48gzm],-(x[sub:24n48gzm]2[/sub:24n48gzm]-6)[sup:24n48gzm]2[/sup:24n48gzm])on parabola (2)

Now find the length of AB and minimize it.

 

[galactus]

Let a point on $\displaystyle y=x^{2}$ be $\displaystyle A(a,a^{2})$

Let a point on $\displaystyle -(x-6)^{2}$ be $\displaystyle B(b,-(b-6)^{2})$

The distance formula is $\displaystyle d=(a-b)^{2}+(a^{2}+(b-6)^{2})^{2}$

Now, set $\displaystyle \frac{{\partial}d}{{\partial}a}=0$ and $\displaystyle \frac{{\partial}d}{{\partial}b}=0$

$\displaystyle \frac{{\partial}d}{{\partial}a}=4a^{3}+2a(2b^{2}-24b+73)-2b=0$........[1]

$\displaystyle \frac{{\partial}d}{{\partial}b}=4b^{3}-72b^{2}+2b(2a^{2}+217)-24a^{2}-2a-864=0$........[2]

Now, can you solve for a and b?.

 

[fatlord]

I'm confused on how you got

= 4a^(3) + 2a(2b^(2) - 24b + 73) - 2b = 0

and

4b^(3) - 72b^(2) + 2b(2a^(2) + 217) -24a^(2) - 2a - 864 = 0

Did you just do the Derivative?

 

[galactus]

Yes, I took the derivatives. See the partial derivative symbol, $\displaystyle {\partial}$?.

 

[fatlord]

OK so i just plug in x^(2) and -(x-6)^(2) for a and b then solve?

 

[Deleted member 4993]

OK so i just plug in x^(2) and -(x-6)^(2) for a and b then solve?

In which equation you are planning this proposed plugging operation?

 

[fatlord]

I'm sorry i'm having difficulty solving for a and b.

 

[Deleted member 4993]

I'm sorry i'm having difficulty solving for a and b.

How far did you go - exactly where you are having trouble?

 

[fatlord]

For the first equation i'm getting

a = - sqrt [( 2b^(2) -26b +73) / 2]

I think i'm doing something wrong

 

[Deleted member 4993]

For the first equation i'm getting

a = - sqrt [( 2b^(2) -26b +73) / 2]<<< By what operations did you arrive at that?

I think i'm doing something wrong

What grade/level of math class are you taking?