A line l intersects the parabola y=x^2 at two points, (a, a^2) and (b, b^2) ba.
1. Calculate the area bordered by the line and the curve:
2. If l is perpendicular to the tangent line to the curve at (a, a^2) Express a in terms of b.
3. When (2.) holds true, find the minimum area m of S in terms of b. For what value of b, the area is minimized?
This is what I got until now http://puu.sh/ph1Dl/9fda67a36e.jpg
mind correcting me?

Question

A line l intersects the parabola y=x^2 at two points, (a, a^2) and (b, b^2) b>a.
1. Calculate the area bordered by the line and the curve: 
2. If l is perpendicular to the tangent line to the curve at (a, a^2) Express a in terms of b.
3. When (2.) holds true, find the minimum area m of S in terms of b. For what value of b, the area is minimized?
This is what I got until now http://puu.sh/ph1Dl/9fda67a36e.jpg
mind correcting me?

loser66 · Answer

To me, for part1)
you have equation of the line passes through A (a, a^2) and B(b,b^2) is
$y_{line}= (a+b)x -ab$
but the area bordered by the line and the curve is 
$$\int_a^b upper - lower = \int_a^b ((a+b)x-ab -x^2)dx$$

loser66 · Answer

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loser66 · Answer

for part 2) the line tangent to the curve has slope = 2x
Hence at (a, a^2) , its slope is $m_1 = 2a$
then perpendicular to the tangent to the curve at (a,a^2) is $m_2= \dfrac{-1}{2a}$

loser66 · Answer

and line from part 1 has the slope is a+b, hence $a+b =\dfrac{-1}{2a}$
you can solve for a.

mathmale · Answer

Interesting problem.  Please, @feynanon, give us a progress report.  How much of this problem have you been able to complete?  What do you still need help with, if anything?

feynanon · Answer

Hello, @Loser66 and @mathmale Thank you so much for your responses! This is how I ended up solving the problem, with some help. Even though I don't know if it's correct, would you mind taking a look at my solution? (excuse my handwriting) 1st: http://puu.sh/ph1Dl/9fda67a36e.jpg 2nd: http://puu.sh/pihEi/14bda90ae2.jpg