document.write( "Question 1200122: Find the point on the ellipse 𝑥^2 + 5𝑦^2 = 36 which is the closest, and which is the farthest point from the line 2𝑥 − 5𝑦 + 30 = 0. Sketch the graph \n" ); document.write( "

Algebra.Com's Answer #834157 by mccravyedwin(407) $\"\"$ $\"About$

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document.write( "We can also do the problem without using calculus:\r\n" );
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document.write( "We draw two lines tangent to the ellipse parallel to the given line.\r\n" );
document.write( "Their points of tangency will be the closest and farthest points\r\n" );
document.write( "on the ellipse from the given line. \r\n" );
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document.write( "We put the given line in slope-intercept form\r\n" );
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document.write( "The green tangent lines will have the same slope with a different\r\n" );
document.write( "y-intercept b\r\n" );
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document.write( "So either green tangent line will have equation \r\n" );
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document.write( "When we solve the system by substitution\r\n" );
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document.write( "There must be one and only one point of intersection.\r\n" );
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document.write( "For there to be only one solution for the point of tangency, the\r\n" );
document.write( "discriminant must = 0\r\n" );
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document.write( "Substituting \r\n" );
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document.write( "x=-4\r\n" );
document.write( "Substituting in the equation of the line:\r\n" );
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document.write( "So one point is (-4,2), that's the closest point.\r\n" );
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document.write( "Substituting \r\n" );
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document.write( "x=4\r\n" );
document.write( "Substituting in the equation of the line:\r\n" );
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document.write( "So the other point is (4,-2), that's the farthest point.\r\n" );
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document.write( "Edwin

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