Question

Show that the line 2x+3y=12 is tangent to the ellipse 4x^2+9y^2=72

Answer 1

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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