Question 331736
1)if the line 3x+4y+k=0 touches the ellipse 9x2 +16y2=144 find the value of k
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The slope of the line is -3/4.
The slope of the ellipse is -9x/16y
Find the 2 points where -9x/16y = -3/4
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-9x/16y = -3/4
y = 3x/4
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Sub for y in {{{9x^2 + 16y^2 = 144}}}
{{{9x^2 + 16(9x^2/16) = 144}}}
{{{18x^2 = 144}}}
x = ± sqrt(8)
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{{{16y^2 = 144 - 72 = 72}}}
y = ± 3sqrt(2)/2
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The 2 points are (2sqrt(2),3sqrt(2)/2) and (-2sqrt(2),-3sqrt(2)/2)
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Use y = mx + b and the points to find b, and then k.
3sqrt(2)/2 = (-3/4)*(2sqrt(2)) + b
b = 3sqrt(2)
y = (-3/4)x + 3sqrt(2)
3x + 4y = 12sqrt(2)
k = -12sqrt(2)
k = +12sqrt(2)