# Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4). - Mathematics and Statistics

Sum

Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4).

#### Solution

Differentiating y =x2 w.r.t. x, we get "dy"/"dx" = "2x"

Slope of tangent at P(2, 4) = ("dy"/"dx")_("at""P"(2,4)) = 2 × 2 = 4

∴ the equation of tangent at P is

y - 4 = 4(x - 2)

∴ y = 4x - 4

∴  y = 4x is equation of line parallel to the tangent at P and passing through the origin O.

4x = y, z = 0

∴ "x"/1 = "y"/4, "z" = 0

∴ the direction ratios of this line are 1, 4, 0

∴ its direction cosines are

+- 1/(sqrt(1^2 + 4^2 + 0^2)), +-4/sqrt(1^2 + 4^2 + 0^2), 0

i.e. +- 1/sqrt17, +-4/sqrt17, 0

∴ unit vectors parallel to tangent line at P(2, 4) is

+- 1/sqrt17(hat"i" + 4hat"j")

Concept: Vectors and Their Types
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