Lesson 1.10 Tangents and Normals (Day 2)

Key Concepts

• Tangent to a curve at point $x = x<em>0$: line with slope $m = \left.\dfrac{dy}{dx}\right|</em>{x<em>0}$ passing through $(x</em>0, y_0)$.

• Normal: line perpendicular to tangent ⇒ slope $m<em>N = -\dfrac{1}{m</em>T}$.

• Tangent from external point $(x<em>P, y</em>P)$ to curve $f(x)$ found by equating slope of line $P(x<em>P, y</em>P)\text{–}T(x, f(x))$ to derivative at $T$.

• For a required slope $m$, solve $\dfrac{dy}{dx} = m$ to find point(s) of contact.

Example Summary

• Example 1

• Curve: $y = 16 - x^2$, line: $y = -x + 4$ intersects at A, B.

• Find intersection; choose A; slope $m = \left.\dfrac{dy}{dx}\right|<em>A = -2x</em>A$.

• Tangent: $y - y<em>A = m(x - x</em>A)$.

• Example 2

• Curve unspecified (implied): solve $\dfrac{dy}{dx} = \text{slope of given line}$.

• Ensure tangent line shares that slope and point of contact.

• Example 3

• External point: $(4,0)$, curve $y^2 = 4ax$ (general parabola form assumed).

• Use equation of chord of contact or solve quadratic obtained from equal‐slope condition to find tangent points.

• Example 4

• Tangent $y = 2x$ touches $y = x^3 + ax + b$ at $x = 1$.

• Conditions:

  1. Contact point lies on curve: $1^3 + a(1) + b = 2(1)$.

  2. Slopes equal: $\left.3x^2 + a\right|_{x=1} = 2$.

• Solve simultaneous equations for $a, b$.

• Example 5

• External point $(2,3)$ to $y = x^2$.

• Let contact point $(t,t^2)$.

• Tangent slope at $t$: $m = 2t$.

• Equation through external point: $3 - t^2 = (2t)(2 - t)$.

• Solve quadratic in $t$ to get both tangent points and equations.

Homework / Practice

• Textbook p.388 Ex 16A: #3, 4, 7, 12, 14, 16.

• Worksheet “Equations of Tangents and Normals” – circled questions only.

• Handout: “More Tangents and Normals”.