Question

Answer the following:

Find the equations of the tangents to the parabola y² = 9x through the point (4, 10).

Answer 1

The equation of the parabola is y² = 9x

Comparing with y² = 4ax, we get,

4a = 9

∴ a = `9/4`

Let m be the slope of the tangent drawn from the point (4, 10) to the parabola.

∴ its equation is

y = `"m"x + "a"/"m"`

∴ y = `"m"x + 9/(4"m")`

∵ (4, 10) lies on it

∴ 10 = `4"m" + 9/(4"m") = (16"m"^2 + 9)/(4"m")`

∴ 40m = 16m² + 9

∴ 16m² – 40m + 9 = 0

∴ 16m² – 4m – 36m + 9 = 0

∴ 4m(4m – 1) – 9(4m – 1) = 0

∴ (4m – 1)(4m – 9) = 0

∴ m = `1/4` or m = `9/4`

Using slope-point form, the equations of tangents are

y – 10 = `1/4(x - 4)` and y – 10 = `9/4(x - 4)`

∴ 4y – 40 = x – 4 and 4y – 40 = 9x – 36 

∴ x – 4y + 36 = 0 and 9x – 4y + 4 = 0.