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Question
Answer the following:
Find the equations of the tangents to the parabola y2 = 9x through the point (4, 10).
Solution
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.
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