Find the equation of the tangent to the ellipse x^2/5 + y^2/4 passing through the point (2, -2).

Question

Find the equation of the tangent to the ellipse \(\frac {x^2}{5} + \frac {y^2}{4} = 1\) passing through the point (2, -2).

x²/5 + y²/4

Answer 1

Given equation of the ellipse is \(\frac {x^2}{5} + \frac {y^2}{4} = 1\)

Comparing this equation with \(\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1\)

we get 

a² = 5 and b² = 4

Equations of tangents to the ellipse \(\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1\) having slope m are

Squaring both the sides, we get 

4m² + 8m + 4 = 5m² + 4 

∴ m² – 8m = 0 

∴ m(m – 8) = 0 

∴ m = 0 or m = 8 

These are the slopes of the required tangents. 

∴ By slope point form y – y₁ = m(x – x₁), 

the equations of the tangents are 

y + 2 = 0(x – 2) and y + 2 = 8(x – 2) 

∴ y + 2 = 0 and y + 2 = 8x – 16 

∴ y + 2 = 0 and 8x – y – 18 = 0.