# At what point on the curve 3x^2 + 2y^2 = 8 does the tangent make an angle of 30 degrees with the x-axis.

Tushar Chandra | Certified Educator

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If the tangent drawn at any point on the curve 3x^2 + 2y^2 = 8 makes an angle of 30 degrees with the x-axis, its slope at that point is tan 30 = `1/sqrt 3`

The slope of a tangent drawn to the curve at point is equal to the value of `dy/dx` at that point.

3x^2 + 2y^2 = 8

`6x + 4y*(dy/dx) = 0`

=> `dy/dx` = `(-6x)/(4y)`

=>` dy/dx` = `(-3x)/(2y)`

If `dy/dx` = `1/sqrt 3`

`1/sqrt 3` = `(-3x)/(2y)`

`-3*x*sqrt 3 = 2*y`

27*x^2 = 4*y^2

As the point lies on the curve 3x^2 + 2y^2 = 8

=> 3x^2 + (27/2)x^2 = 8

=> (33/2)x^2 = 8

=> 33x^2 = 16

=> x =` 4/sqrt 33`

y =` sqrt((8 - 3*16/33)/2)`

=> -`sqrt(36/11)`

The point on the curve 3x^2 + 2y^2 = 8 where the tangent makes an angle 30 degrees with the x-axis is` (4/sqrt 33, -6/sqrt11)`

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