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 #834154 by Edwin McCravy(20059)\"\" \"About 
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document.write( "We draw two lines tangent to the ellipse parallel to the given line.\r\n" );
document.write( "Their point 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( "\"x%5E2+%2B+5y%5E2+=+36\"\r\n" );
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document.write( "Differentiate implicitly:\r\n" );
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document.write( "\"2x+%2B+10y%2Aexpr%28dy%2Fdx%29=0\"\r\n" );
document.write( "Solve for \"expr%28dy%2Fdx%29\"\r\n" );
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document.write( "\"dy%2Fdx=-x%2F%285y%29\"\r\n" );
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document.write( "We find the slope of the line:\r\n" );
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document.write( "\"2x+%E2%88%92+5y+%2B+30+=+0\"\r\n" );
document.write( "\"y=expr%282%2F5%29x%2B6\"\r\n" );
document.write( "\"slope=2%2F5\"\r\n" );
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document.write( "Find the points on the ellipse where the slope of the tangent lines\r\n" );
document.write( "are parallel to the given line by setting the derivative of ellipse\r\n" );
document.write( "equal to slope of line: \r\n" );
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document.write( "\"x%2F%285y%29=2%2F5\"\r\n" );
document.write( "solve for x\r\n" );
document.write( "\"x=-2y\" \r\n" );
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document.write( "Substitute in ellipse's equation\r\n" );
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document.write( "\"x%5E2+%2B+5y%5E2+=+36\"\r\n" );
document.write( "\"%28-2y%29%5E2%2B5y%5E2=36\"\r\n" );
document.write( "\"4y%5E2%2B5y%5E2=36\"\r\n" );
document.write( "\"9y%5E2=36\"\r\n" );
document.write( "\"y%5E2=4\"\r\n" );
document.write( "\"y=%22%22+%2B-+2\"\r\n" );
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document.write( "\"x%5E2+%2B+5y%5E2+=+36\"\r\n" );
document.write( "\"x%5E2+%2B+5%28%22%22+%2B-+2%29%5E2=36\"\r\n" );
document.write( "\"x%5E2+%2B+5%284%29=36\"\r\n" );
document.write( "\"x%5E2+%2B20=36\"\r\n" );
document.write( "\"x%5E2=16\"\r\n" );
document.write( "\"x=%22%22+%2B-+4\"\r\n" );
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document.write( "Four points on the ellipse are (-4,2), (-4,-2), (4,2), (4,-2)\r\n" );
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document.write( "The only 2 points on the green tangent lines above are (-4,2) and (4,-2).\r\n" );
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document.write( "They are the required points.\r\n" );
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document.write( "Edwin
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