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Let $a,b$ be real positive numbers such that $b>a$.If the area of the region that between the two lines $ax+by=20$,$ax+by=30$ and the positive part of the axes $X,Y$ is equal to $10$ unit square.How to find $a\cdot b$

jimjim
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user64461
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1 Answers1

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Hint:

It can be done with integrals, but who has time? Since the given lines are parallel, the wanted region is an isosceles trapezoid with bases of lengths $\,\displaystyle{20\sqrt{\frac{a^2+b^2}{a^2b^2}}=\frac{20\sqrt{a^2+b^2}}{ab}\;,\;30\sqrt{\frac{a^2+b^2}{a^2b^2}}=\frac{30\sqrt{a^2+b^2}}{ab}}\,$ .

I'll leave it to you to calculate the trapezoid's height and thus its area (further hint: use the formula for distance from a point to a straight line...)

It is assumed above that $\,a>0\,$ .

Added on request: The distance between the point $\,(x_0,y_0)\,$ and the straight line $\,Ax+By+C=0\,$ is given by

$$\frac{\left|Ax_0+By_0+C\right|}{\sqrt{A^2+B^2}}$$

DonAntonio
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  • Can you tell what is the formula ? – user64461 Mar 01 '13 at 04:40
  • @user64461 , I added on my answer the formula. – DonAntonio Mar 01 '13 at 07:40
  • Sorry to bother you but i think i need further help.How is this formula can help me to find the height ? What do i have now to use ? – user64461 Mar 01 '13 at 07:48
  • Did you draw the two given lines? Did you notice these are parallel lines and they conform the bases of an (isosceles) trapezoid formed by the two lines and the two axis in the first quadrant? Then, as basic geometry shows, the trapezoid's height is the distance between both bases, which in this case is the distance between any point on one of the lines to the other line! After you have the trapezoid's height, and since you can easily find what the bases' lengths are, use the known formula for the trapezoid's area. You didn't write anything but I assume you know some analytic geometry... – DonAntonio Mar 01 '13 at 07:54
  • Apparently somebody disliked my answer....oh, well. – DonAntonio Mar 01 '13 at 08:48
  • Not me of course but i could not determine the answer from your hints until now . – user64461 Mar 01 '13 at 08:51
  • Well @user64461 , you can always check the great hint given by Steve Kass. Anyway, the answer is $,ab=25,$... – DonAntonio Mar 01 '13 at 08:55
  • Are you sure ? may be it is $13$ – user64461 Mar 01 '13 at 09:33