Find the equation of the tangent to the ellipse x2 + 4y2 = 9 which are parallel to the line 2x + 3y – 5 = 0. - Mathematics and Statistics

Sum

Find the equation of the tangent to the ellipse x2 + 4y2 = 9 which are parallel to the line 2x + 3y – 5 = 0.

Solution

We know that the equations of tangents with slope m to the ellipse x^2/"a"^2 + y^2/"b"^2 = 1 are

y = "m"x ± sqrt("a"^2"m"^2 + "b"^2)   ...(1)

The equation of the ellipse is x2 + 4y2 = 9

∴ x^2/9 + y^2/((9/4) = 1

Comparmg this with x^2/"a"^2 + y^2/"b"^2 = 1, we get

a2 = 9, b2 = 9/4

Slope of 2x + 3y – 5 = 0 is -2/3

The required tangent is parallel to it

∴ its slope = m = -2/3

Using (1), the required equations of tangents are

y = -(2x)/3 ± sqrt(9 xx 4/9 + 9/4)

∴ y = -(2x)/3 ± sqrt(25/4)

∴ y = -(2x)/3 ± 5/2

∴ 6y = – 4x ± 15

∴ 4x + 6y = ± 15

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