Parametric forms of all conics — Hinglish notes

3.4.12 · Maths › Conic Sections

Parametric form KYA hota hai?

Yahan magic tool ek trig / hyperbolic identity hota hai jiska shape conic ke equation se match kare.

Identity	Conic se match
$cos^{2} θ + sin^{2} θ = 1$	ellipse / circle
$sec^{2} θ - tan^{2} θ = 1$	hyperbola
$y^{2} = x \cdot (\dots)$ direct	parabola

Har ek ko (scratch se) DERIVE kaise karte hain

1. Circle $x^{2} + y^{2} = r^{2}$

Hume $x, y$ chahiye jisse $x^{2} + y^{2}$ automatically $r^{2}$ ban jaaye. Identity $cos^{2} θ + sin^{2} θ = 1$ bilkul perfect lagti hai. Use $r^{2}$ se multiply karo: $r^{2} cos^{2} θ + r^{2} sin^{2} θ = r^{2} .$ $x^{2} + y^{2} = r^{2}$ se compare karo: term by term match karo.

2. Ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$

Cartesian equation already " $(\cdot)^{2} + (\cdot)^{2} = 1$ " form mein hai. Toh set karo $\frac{x}{a} = cos θ, \frac{y}{b} = sin θ .$ Ye step kyon? Tab $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = cos^{2} θ + sin^{2} θ = 1$ — identity saara kaam kar deti hai.

3. Hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$

Ab yahan minus hai. Hume ek aise identity ki zaroorat hai jisme $(\cdot)^{2} - (\cdot)^{2} = 1$ ho. Exactly yahi hai $sec^{2} θ - tan^{2} θ = 1$ . Set karo $\frac{x}{a} = sec θ, \frac{y}{b} = tan θ .$

4. Parabola $y^{2} = 4 a x$

Yahan koi sum/difference identity ki zaroorat nahi — equation directly $x$ aur $y^{2}$ ko link karti hai. $y$ ko parameter $t$ ke proportional karo: $y = 2 a t$ rakho. Tab $y^{2} = 4 a^{2} t^{2} = 4 a x ⟹ x = a t^{2} .$ $y = 2 a t$ kyon, $y = t$ kyon nahi? Humne neat constant $2 a$ choose kiya taki $x$ clean $a t^{2}$ aaye. Koi bhi choice chalti hai; yahi standard "nice" wali hai.

Worked examples

Common mistakes (steel-manned)

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek curve ek racetrack hai. Har point ko "kitna right aur kitna upar" se describe karne ki jagah, hum ek stopwatch time $t$ dete hain. Har time par car ek definite jagah hoti hai. Agar main kahoon "time $t = 3$ ", aap turant jaante ho car kahaan hai. Circle ke liye stopwatch angle hai jitna tum ghoom gaye. Parabola ke liye bas ek plain number hai. Ek number poori position bata deta hai — yahi parametric form hai.

Summary table

Conic	Cartesian	Parametric	Identity used	Range
Circle	$x^{2} + y^{2} = r^{2}$	$(r cos θ, r sin θ)$	$c^{2} + s^{2} = 1$	$[0, 2 π)$
Ellipse	$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$	$(a cos θ, b sin θ)$	$c^{2} + s^{2} = 1$	$[0, 2 π)$
Parabola	$y^{2} = 4 a x$	$(a t^{2}, 2 a t)$	direct	$R$
Hyperbola	$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$	$(a sec θ, b tan θ)$	$sec^{2} - tan^{2} = 1$	$θ \neq = \frac{π}{2}$
Hyperbola (alt)	same	$(a cosh t, b sinh t)$	$cosh^{2} - sinh^{2} = 1$	right branch

Connections

Conic Sections - Standard equations
Eccentric angle and auxiliary circle
Tangents and normals to conics
Chord of contact and pole-polar
Trigonometric identities
Hyperbolic functions
Parametric differentiation

#flashcards/maths

Circle

x^{2} + y^{2} = r^{2}

ki parametric form kya hai?

x = r cos θ, y = r sin θ

Ellipse parametrization ko kaunsi identity power karti hai?

cos^{2} θ + sin^{2} θ = 1

Ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

ki parametric form

x = a cos θ, y = b sin θ

(a cos θ, b sin θ)

hyperbola ko parametrize kyon nahi kar sakta?

Hyperbola mein minus sign hota hai;

sec^{2} - tan^{2} = 1

chahiye,

cos^{2} + sin^{2}

nahi

Hyperbola

\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1

ki parametric form (trig)

x = a sec θ, y = b tan θ

Hyperbola ki alternative hyperbolic parametrization

x = a cosh t, y = b sinh t

(right branch)

Parabola

y^{2} = 4 a x

ki parametric form

x = a t^{2}, y = 2 a t

Ellipse form mein

θ

ko kya kehte hain (geometric angle nahi)?

Eccentric angle

y^{2} = 4 a x

par parameter

t

par tangent

t y = x + a t^{2}

(a t^{2}, 2 a t)

se parabola ka slope at point

t

1/ t

Parametric form verify kaise karte hain?

Parameter eliminate karo aur Cartesian equation identically recover karo