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Fill in the blank below. The shortest and least-curved arc between two points on a surface is a The shortest and least-curved arc between two points on a surface is a sphere. plane. geodesic. pseudosphere. line. 

          Fill in the blank below.
The shortest and least-curved arc between two points on a surface is a
The shortest and least-curved arc between two points on a surface is a
sphere.
plane.
geodesic.
pseudosphere.
line.
Fill in the blank below. The shortest and least-curved arc between two points on a surface is a The shortest and least-curved arc between two points on a surface is a sphere. plane. geodesic. pseudosphere. line.

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Elementary and Intermediate Algebra
Elementary and Intermediate Algebra
Alan S. Tussy, R. David Gustafson 5th Edition
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Transcript

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00:01 Hello students, here we have given points that is x1, y1 and x2, y2 and the path y is equal to y of x.
00:17 Now to find the path y of x that has the shortest length that is in a straight line, we consider length of the path is equal to dx square plus dy square power 1 by 2 that is root.
00:38 Now dy is equal to dy by dx that is y dash of x into dx.
00:47 Now substituting this in the above equation, ds is equal to 1 plus y dash of x square power 1 by 2 as it is.
01:02 Here we take common as dx from both, hence we write dx because dx square power 1 by 2 is dx.
01:14 The shortest distance of the path between points 1 and 2 is given by this equation.
01:21 Now in polar coordinates, we consider x is equal to r cos theta, y is equal to r sin theta, then dx is equal to r minus sin theta d theta and dy is equal to r cos theta d theta.
01:49 Hence, we can write y dash of x is equal to dy by dx dividing r cos theta by r sin theta, rr get cancelled and we get minus cot theta because cos theta by sin theta is cot theta.
02:13 Hence we obtain the equation.
02:16 Next to calculate distance, let us consider starting from a and ending to b at t is equal to 1...
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