Show that the line 2x+3y=12 is tangent to the ellipse 4x^2+9y^2=72
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Given : line 2x+3y=12 is tangent to the ellipse 4x²+9y²=72
To Find : Prove
Solution:
4x²+9y²=72
=> 8x + 18y dy/dx = 0
=> dy/dx = - 8x/18y
=> dy/dx = - 4x/9y
2x+3y=12
Squaring both sides
=> 4x²+9y² + 12xy = 144
4x²+9y²=72
=> 12xy = 72
=> xy = 6
2x+3y=12
xy = 6
=> 2x + 3(6/x) = 12
=> x² + 9 - 6x = 0
=> ( x - 3)² = 0
=> x = 3 , y = 2
on solving x = 3 , y = 2
dy/dx = - 4x/9y = - 4 * 3 / ( 9 * 2) = - 2/3
2x+3y=12 => y = -2x/3 + 4
Hence slope = 2/3
So line 2x+3y=12 is tangent to the ellipse 4x²+9y²=72
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