Question

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

Answer 1

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")`