Q.283. Equations of tangents to the ellipse from an external point.

Question 283.
Find the equation of both tangent lines to the ellipse x^2 + 4y^2 = 36 that pass through the point (5,3) which is not a point on the ellipse.

Answer 283.
x^2 + 4y^2 = 36
=> x^2/36 + y^2/9 = 1
Comparing with x^2/a^2 + y^2/b^2 = 1,
a^2 = 36 and b^2 = 4

The tangent to this ellipse with slope m has equation of the form,
y = mx ± √(a^2m^2 + b^2)
=> y = mx ± √(36m^2 + 9)

If it passes through (5, 3),
3 = 5m ± √(36m^2 + 9)
=> (3 - 5m)^2 = 36m^2 + 9
=> 11m^2 + 30m = 0
=> m(11m + 30) = 0
=> m = 0 or - 30/11

=> the equations of the two tangents are
y - 3 = 0 (x - 5) or y - 3 = - (30/11) (x - 5)
=> y = 3 OR 30x + 11y = 183.

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