Tangents

Tangents

1. Geometric Meaning of dy/dx

For a curve $y = f(x)$, the derivative at $x = x_0$ is the slope of the tangent at $P(x_0, y_0)$:

The tangent is the limiting position of the chord $PQ$ as $Q → P$ along the curve. Where $dy/dx = 0$ the tangent is horizontal; where $dy/dx → ∞$ (or $dx/dy = 0$) it is vertical.

2. Equation of the Tangent

If $P(x_0, y_0)$ lies on the curve and $m = f'(x_0)$ exists:

is the point-slope form of the tangent. For an implicitly defined curve $F(x,y)=0$ with $F_x, F_y$ partials at $P$, $m = -F_x/F_y$ (provided $F_y ≠ 0$), so $F_x(x_0,y_0)(x-x_0)+F_y(x_0,y_0)(y-y_0)=0$ is same line.

• Parametric form $x = x(t), y = y(t)$:

• Polar form $r = r(θ)$: slope in Cartesian uses $x = r cosθ, y = r sinθ$, giving $dy/dx = [r' sinθ + r cosθ]/[r' cosθ - r sinθ]$.

3. Tangents Parallel / Perpendicular to Axes

Condition on $m$	Geometric position
$m = 0$ ($dy/dx = 0$)	Tangent ∥ x-axis (horizontal)
$m → ∞$ ($dx/dy = 0$)	Tangent ∥ y-axis (vertical)
$m = 1$ or $m = -1$	Inclined at $45°$ or $135°$

Finding such points is a standard exam question: solve $f'(x)=0$ for horizontals, or where denominator zero for verticals.

4. Angle of Intersection of Two Curves

If two curves meet at $P$ with tangent slopes $m_1, m_2$, the acute angle $φ$ between them is:

• $m_1 = m_2$ ⇒ curves touch (common tangent, $φ=0$).
• $m_1 m_2 = -1$ ⇒ curves cut orthogonally ($φ = 90°$, $tan φ → ∞$).

5. Worked Examples

(a) Curve y = x^3 - 3x^2 + 2 at P(1,0):
    dy/dx = 3x^2 - 6x  ⇒  m = 3 - 6 = -3
    Tangent: y - 0 = -3(x - 1)  ⇒  y + 3x - 3 = 0               ✓

(b) Parametric: x = a cos^3 t , y = a sin^3 t (astroid) at t = π/4:
    dx/dt = -3a cos^2 t sin t , dy/dt = 3a sin^2 t cos t
    m = (3a sin^2 t cos t)/(-3a cos^2 t sin t) = -tan t = -1 at π/4
    Point: (a/2√2, a/2√2) ; Tangent: y - a/2√2 = -1(x - a/2√2)    ✓

(c) Implicit: x^2 + y^2 = 25 at (3,4):
    2x + 2y y' = 0 ⇒ y' = -x/y = -3/4
    Tangent: 4y -16 = -3x +9 ⇒ 3x+4y-25=0                         ✓

(d) Orthogonal check: y^2 = 4ax (parabola) and x^2 + y^2 = 8ax at intersection
    For y^2=4ax : 2y y' =4a ⇒ y' =2a/y = √(a/x) ; at common point solve → m1
    For circle: 2x+2y y'=8a ⇒ y' =(4a -x)/y ; product m1·m2 = -1 ⇒ orthogonal ✓

6. Subtangent and Subnormal (often asked)

For $P(x_0,y_0)$ with $m = y'_0$, projection on x-axis:

• Subtangent $= y_0 / m = y_0 · dx/dy$
• Subnormal $= y_0 · m = y_0 · dy/dx$

Triangle $PTN$ where $T$ = foot of tangent on x-axis, $N$ = foot of normal.

7. Exam Points

• Derive $m = dy/dx$ as slope and write $y - y_0 = m(x-x_0)$ — 3 marks, almost certain.
• Find points where tangent is parallel to x-axis / y-axis — 5 marks, very frequent.
• Equation of tangent to parametric/polar or implicit curve at given point — 5 marks, very frequent.
• Angle of intersection / condition for orthogonal intersection ($m_1 m_2 = -1$) — 5 marks, frequent.
• Subtangent / subnormal length — 3-5 marks.