SOLUTION: an ellipse is defined by {{{x^2/81 + y^2/64 = 1}}}. Find the equations of the lines tangent to this ellipse which make and angel of 45 degrees with the x-axis can someone

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: an ellipse is defined by {{{x^2/81 + y^2/64 = 1}}}. Find the equations of the lines tangent to this ellipse which make and angel of 45 degrees with the x-axis can someone       Log On


   

Question 1016362: an ellipse is defined by x%5E2%2F81+%2B+y%5E2%2F64+=+1. Find the equations of the lines tangent to this ellipse which make and angel of 45 degrees with the x-axis



can someone show me how to do it and answer it. thankyou very much for the help
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Differentiate,
%282xdx%29%2F81%2B%282ydy%29%2F64=0
%28ydy%29%2F64=-%28xdx%29%2F81
dy%2Fdx=-%28x%2Fy%29%2864%2F81%29
The slope of the tangent line is equal to the value of the derivative.
An angle of 45 degrees is equivalent to a slope of 1.
1=-%28x%2Fy%29%2881%2F64%29
y=-%2864%2F81%29x
This point also satisfies the ellipse equation,
x%5E2%2F81%2B%281%2F64%29%2864%2F81%29%5E2x%5E2=1
%281%2F81%2B64%2F81%5E2%29x%5E2=1
%28145%2F6561%29x%5E2=1
x%5E2=6561%2F145
x=0+%2B-+81%2Fsqrt%28145%29
x=0+%2B-+%2881%2F145%29sqrt%28145%29
Then plugging that into the ellipse equation you get,
y=+0+%2B-+%2864%29%2Fsqrt%28145%29
y=0+%2B-+%2864%2F145%29sqrt%28145%29
So the two points are,
(%2881%2F145%29sqrt%28145%29,-%2864%2F145%29sqrt%28145%29)
(-%2881%2F145%29sqrt%28145%29,%2864%2F145%29sqrt%28145%29)
That's when the slope is 1.
Similarly when the slope is -1.
(%2881%2F145%29sqrt%28145%29,%2864%2F145%29sqrt%28145%29)
(-%2881%2F145%29sqrt%28145%29,-%2864%2F145%29sqrt%28145%29)
Use the point slope form of a line to get the equation of the tangent line.
y%2B64%2Fsqrt%28145%29=1%28x-81%2Fsqrt%28145%29%29
y=x-145%2Fsqrt%28145%29
y=x-sqrt%28145%29
Similarly,
y=x%2Bsqrt%28145%29
y=-x%2Bsqrt%28145%29
y=-x-sqrt%28145%29
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