Normal of Algebraic

1. From Tangent to Normal

The normal at $P(x_0,y_0)$ is the line through $P$ perpendicular to the tangent. If tangent slope is $m_T = [dy/dx]_P$ (finite and non-zero), normal slope is:

m_N = -1/m_T = -[dx/dy]_P

Hence, for $m_T ≠ 0, ∞$ :

y - y_0 = -1/m_T (x - x_0)

is the equation of the normal. Special cases: if tangent is horizontal ( $m_T=0$ ) the normal is vertical $x = x_0$ ; if tangent is vertical ( $m_T→∞$ ) the normal is horizontal $y = y_0$ .

For $F(x,y)=0$ , $F_x(x_0,y_0)(y-y_0) - F_y(x_0,y_0)(x-x_0)=0$ is the normal (swap and sign change vs tangent). Parametric: $m_N = -[dx/dt]/[dy/dt]$ .

2. Algebraic Curves — What Examiners Mean

"Algebraic" signals a polynomial equation $F(x,y)=0$ (circle, ellipse, folium, lemniscate). Implicit differentiation gives $y'$ without solving for $y$ . The normal formula above applies directly to such curves at any non-singular point ( $F_x, F_y$ not both zero). A singular point (both partials zero) has no unique tangent/normal — cusp/self-intersection.

3. Lengths of Tangent, Normal, Subtangent, Subnormal

For $P(x_0,y_0)$ with $m = y'_0$ , let $T$ = meet of tangent with x-axis, $N$ = meet of normal with x-axis, foot $M(x_0,0)$ :

Subtangent MT = |y_0 / m| = |y_0 · dx/dy|  ; Tangent length PT = |y_0|·√(1+1/m²) = |y_0|·√(1+(dx/dy)²)
Subnormal  MN = |y_0 · m| = |y_0 · dy/dx| ; Normal length PN = |y_0|·√(1+m²)

These four lengths form right triangles $PMT$ and $PMN$ with $PM = |y_0|$ .

4. Worked Examples

(a) y = x^3 at (1,1): m_T =3x^2 =3 ⇒ m_N = -1/3
    Normal: y-1 = -1/3(x-1) ⇒ x+3y-4=0                          ✓

(b) Circle x^2+y^2=25 at (3,4): m_T=-x/y=-3/4 ⇒ m_N=4/3
    Normal: y-4=4/3(x-3) ⇒ 4x-3y=0, which passes through origin (as every circle normal does) ✓

(c) Ellipse x^2/25 + y^2/9 =1 at (5cosθ,3sinθ):
    m_T = -(9x)/(25y) = -(3cosθ)/(5sinθ) ; m_N = (5sinθ)/(3cosθ)
    Normal: (5x/cosθ) - (3y/sinθ) =16  (standard form)          ✓

(d) Subnormal: y^2=4ax at (at^2,2at): m=1/t
    Subnormal = y·m = 2at·(1/t)=2a (constant!) ⇒ parabola property ✓

(e) Find points on y = x^3 -3x where normal is parallel to x-axis:
    Normal ∥ x-axis ⇒ tangent ∥ y-axis ⇒ m_T→∞ ⇒ dx/dy=0 ⇒ 3x^2-3=0 ⇒ x=±1
    Points (1,-2) and (-1,2)                                       ✓

5. Normal vs Tangent — Quick Compare

Feature	Tangent at $P$	Normal at $P$
Direction	along curve ( $m_T$ )	perpendicular ( $-1/m_T$ )
Equation	$(y-y_0)=m_T(x-x_0)$	$(y-y_0)=-1/m_T(x-x_0)$
Use	slope, touching line, approximation $Δy≈m_TΔx$	curvature centre lies on normal, evolute

6. Exam Points

Write tangent and normal at $P$ for explicit, implicit (algebraic), parametric curves — 5 marks, very frequent.
Show normal to circle/central conic passes through centre / property — 3-5 marks.
Find points where tangent/normal is parallel to an axis — 5 marks, frequent.
Compute subtangent, subnormal, lengths of tangent/normal — 3-5 marks, frequent.
Condition for two curves to intersect orthogonally — relate to $m_T1·m_T2=-1$ vs normals — 5 marks.

Normal of Algebraic