Tangents & Normals

• A tangent is a straight line that touches a curve at a single point, known as the point of contact, having the same gradient as the curve at that point.

• A normal is a straight line that is perpendicular to the tangent at the point of contact, meaning its gradient is the negative reciprocal of the tangent's gradient.

• In coordinate geometry, these lines are typically defined at a general point using parametric coordinates, such as $(at^2, 2at)$ for a parabola or $(ct, \frac{c}{t})$ for a rectangular hyperbola.

• The relationship between the gradients of the tangent ($m_t$) and the normal ($m_n$) is defined by the perpendicularity condition: $m_t \cdot m_n = -1$.

• The gradient of a curve at any point is found by calculating the first derivative, $\frac{dy}{dx}$, using either Cartesian or parametric differentiation.

• For parametric equations $x=f(t)$ and $y=g(t)$, the gradient is determined by the chain rule: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

• Once the gradient $m$ is known, the equation of the line is constructed using the point-slope form: $y - y_1 = m(x - x_1)$.

• This principle ensures that the tangent accurately represents the instantaneous rate of change of the curve at the specific parameter value $t$.

Step-by-Step Derivation

• Step 1: Identify Coordinates: Express the point of interest in terms of the parameter $t$ (e.g., $x = at^2, y = 2at$).

• Step 2: Differentiate: Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$, then divide them to find the gradient of the tangent, $m = \frac{1}{t}$ for a parabola or $m = -\frac{1}{t^2}$ for a hyperbola.

• Step 3: Find Normal Gradient: Calculate the negative reciprocal of the tangent gradient ($m_{normal} = -\frac{1}{m_{tangent}}$).

• Step 4: Formulate Equation: Substitute the point and the chosen gradient into $y - y_1 = m(x - x_1)$ and simplify to the required form (Cartesian or standard linear form).

• Intersection Problems: To find where two tangents or normals meet, set their equations equal to each other and solve for $x$ or $y$, often resulting in expressions involving multiple parameters (e.g., $p$ and $q$).

• Note that the parabola's tangent gradient depends linearly on the inverse of $t$, while the hyperbola's tangent gradient depends on the inverse square of $t$ and is always negative (for $x, y > 0$).

• Sign Errors: A common mistake is forgetting the negative sign when calculating the gradient of the tangent to a rectangular hyperbola ($-\frac{1}{t^2}$).

• Reciprocal Confusion: Students often mix up the gradients of the tangent and normal, especially when the tangent gradient is already a fraction like $\frac{1}{t}$.

• Parameter Misuse: Using the same parameter $t$ for two different points on the same curve; always use distinct letters like $p$ and $q$ for distinct points.

• Implicit Differentiation Errors: When using Cartesian forms like $y^2 = 4ax$, students often forget to include the $\frac{dy}{dx}$ term when differentiating $y^2$ with respect to $x$.