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Question
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")`
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