find the shortest distance between y = x^2 and y – x + 2 = 0
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Have you learned multi-variable calculus?Hi guys, can anyone explain how to find the shortest distance between the given curve and line.
y = x2 and y – x + 2 = 0
Thanks …
The most efficient solution will require that.
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You (should) know that the shortest distance from a straight line to anything else is going to be along a perpendicular from that straight line. So what must be the slope of that shortest-distance line? And then what will be the slope of the tangent to the parabola where that shortest-distance line meets the parabola? Where does this lead?
If you get stuck, please reply with a clear listing of your thoughts and efforts so far (such as drawing a picture). Thank you!
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Recall the distance from a point, \(\displaystyle (x_{0},y_{0})\), to a line, \(\displaystyle ax+by+c = 0\). \(\displaystyle \dfrac{|ax_{0} + by_{0} + c|}{\sqrt{a^{2}+b^{2}}}\).
Generalizing for the curve, \(\displaystyle y = x^2\), we have \(\displaystyle \dfrac{|-x + x^{2} + 2|}{\sqrt{a^{2}+b^{2}}}\).
Satisfy yourself that \(\displaystyle x^{2} - x + 2 > 0\) and minimize that expression.
Please read my signature. You now have three ways to proceed.
Generalizing for the curve, \(\displaystyle y = x^2\), we have \(\displaystyle \dfrac{|-x + x^{2} + 2|}{\sqrt{a^{2}+b^{2}}}\).
Satisfy yourself that \(\displaystyle x^{2} - x + 2 > 0\) and minimize that expression.
Please read my signature. You now have three ways to proceed.
You (should) know that the shortest distance from a straight line to anything else is going to be along a perpendicular from that straight line. So what must be the slope of that shortest-distance line? And then what will be the slope of the tangent to the parabola where that shortest-distance line meets the parabola? Where does this lead?
If you get stuck, please reply with a clear listing of your thoughts and efforts so far (such as drawing a picture). Thank you!
Unfortunately, i have a weak background in this chapter of calculus, and after watching some videos on youtube i reached this point in the pic which at the end there is a point (0.25, 1.75) only if y was negative.
Finally, can you please correct my mistakes and guide me to websites or youtube channels which helps in kinda problems.
