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Minimum triangle area

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A variable line through some point P(j, k) intersects the x and y axis at points X(a, 0) and Y(0, b). You are to find the minimum are of the triangle XOY.


After a few pages of scratch paper, I gave up and looked up the solution. I got as far as determining that the slope of the variable line would determine the area of the triangle so I used the 'point-slope' equation for the line.



What I don't understand is why is equating x and y to zero in the equation of the line valid? The way I thought about it is when x (or y) is zero, then the line would sort of shoot off to the right and create a line that goes through the Origin and the point P(j, k) which would be an invalid case for the triangle (not being a triangle at all).

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I thought the presenter explained quite well.


She said that you have Area as a function of two variables, x and y.


You can't differentiate this to find a minimum, so you introduce a second equation connecting ,x, y, and a single variable or parameter m.


This second equation is the equation of the line through the point 8,4.


Now we know the values of x or y for two other points on that line. That is when x=0 or y=0.


So we can substitute this (she did this at 3 mins 15 seconds) to find twio equations, one connecting only x and m and the other connecting only y and m.


Then you can substitute for x and y in the original equation obtaining an equation connecting only Area and m.


This you can differentiate to find a minimum.


Does this help?

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Moderator Note



There is no need for comments like that - this forum is open to, and in fact is enriched by, members of all abilities. Our mathematicians range from those who teach at University level to those who are just starting off on their long journey of understanding - we are all here because we enjoy, are intrigued by, or frustrated by mathematics and all are equally welcome.


Please do not denigrate or mock another member's ability. Do not respond to this moderation within the thread



After this video u still need explation?????????????


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