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Find minimum distance between parabolas x^2 and -(x - 6)^2
- Thread starter fatlord
- Start date
D
Deleted member 4993
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fatlord said: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.
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?.
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?.
D
Deleted member 4993
Guest
fatlord said: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?
D
Deleted member 4993
Guest
fatlord said:I'm sorry i'm having difficulty solving for a and b.
How far did you go - exactly where you are having trouble?
D
Deleted member 4993
Guest
fatlord said: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?