General form of circle — converting, finding centre and radius

2.3.14 · Maths › Coordinate Geometry

Overview

Circle ki general form hai $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . Yeh standard form $(x - h)^{2} + (y - k)^{2} = r^{2}$ ke comparison mein thodi messy lagti hai, lekin algebraic manipulation ke baad circles aksar isi roop mein milti hain. Iska asli power hai conversion mein: un linear terms ke andar chupi hui centre aur radius ko nikalna.

[!intuition] General Form Exist Kyun Karti Hai?

Jab tum $(x - h)^{2} + (y - k)^{2} = r^{2}$ ko expand karte ho, toh milta hai: $x^{2} - 2 h x + h^{2} + y^{2} - 2 k y + k^{2} = r^{2}$ $x^{2} + y^{2} - 2 h x - 2 k y + (h^{2} + k^{2} - r^{2}) = 0$

Squared terms ka coefficient 1 hota hai, linear terms centre ko chupaate hain, aur constant term mein centre aur radius dono chuppe hote hain. Real-world problems (loci, intersections, tangents) mein aksar general form seedhi milti hai. Tumhe centre aur radius ko reverse-engineer karna padta hai.

[!definition] General Form ke Components

$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$

$x^{2}$ aur $y^{2}$ ke Coefficients: Dono 1 hone chahiye (agar nahi hain, toh divide kar lo)
Koi $x y$ term nahi: Cross-product coefficient 0 hona chahiye (pure circle, ellipse/hyperbola nahi)
$2 g$ : $x$ ka coefficient → $- 2 h$ encode karta hai, jahan $h$ , centre ka $x$ -coordinate hai
$2 f$ : $y$ ka coefficient → $- 2 k$ encode karta hai, jahan $k$ , centre ka $y$ -coordinate hai
$c$ : Constant → $h^{2} + k^{2} - r^{2}$ encode karta hai

[!formula] Conversion Formulas (Scratch se Derive Kiye Hue)

Goal: $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ se centre $(- g, - f)$ aur radius $r$ nikalna.

Derivation: Completing the Square

Yeh technique kyun? Kyunki $(x - h)^{2}$ linear $x$ term ko vanish kar deta hai, jisse $h$ isolate ho jaata hai.

Step 1: $x$ terms aur $y$ terms ko group karo: $(x^{2} + 2 g x) + (y^{2} + 2 f y) + c = 0$

Step 2: $x$ ke liye square complete karo:

$x$ ke coefficient ka aadha lo: $\frac{2 g}{2} = g$
Use square karo: $g^{2}$
Add aur subtract karo: $x^{2} + 2 g x + g^{2} - g^{2}$

$x^{2} + 2 g x = (x + g)^{2} - g^{2}$

Kyun? Kyunki $(x + g)^{2} = x^{2} + 2 g x + g^{2}$ . Hum $g^{2}$ "borrow" karke perfect square banate hain.

Step 3: Isi tarah $y$ ke liye square complete karo: $y^{2} + 2 f y = (y + f)^{2} - f^{2}$

Step 4: Wapis substitute karo: $(x + g)^{2} - g^{2} + (y + f)^{2} - f^{2} + c = 0$ $(x + g)^{2} + (y + f)^{2} = g^{2} + f^{2} - c$

Step 5: $(x - h)^{2} + (y - k)^{2} = r^{2}$ se compare karo:

$(x + g)^{2} = (x - (- g))^{2} ⟹ h = - g$
$(y + f)^{2} = (y - (- f))^{2} ⟹ k = - f$
$r^{2} = g^{2} + f^{2} - c$

Final Result

Negative signs kyun? General form $+ 2 g x$ likhti hai, lekin standard form mein $- 2 h x$ hota hai. Toh $2 g = - 2 h ⟹ g = - h$ .

[!example] Example 1: Standard Conversion

Problem: $x^{2} + y^{2} + 6 x - 4 y - 12 = 0$ ka centre aur radius nikalo.

Solution:

Step 1: $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ se match karke coefficients identify karo:

$2 g = 6 ⟹ g = 3$
$2 f = - 4 ⟹ f = - 2$
$c = - 12$

Yeh step kyun? Direct pattern matching se parameters isolate ho jaate hain.

Step 2: Formulas apply karo:

Centre: $C = (- g, - f) = (- 3, - (- 2)) = (- 3, 2)$
Radius: $r = g^{2} + f^{2} - c = 9 + 4 - (- 12) = 25 = 5$

$g^{2} + f^{2} - c$ check kyun karein? Agar negative ho, toh "circle" imaginary hai (koi real point ise satisfy nahi karta).

Answer: Centre $(- 3, 2)$ , radius $5$ .

[!example] Example 2: Standard se General mein Convert Karna

Problem: $(x - 2)^{2} + (y + 5)^{2} = 16$ ko general form mein convert karo.

Solution:

Step 1: Squares expand karo: $(x - 2)^{2} = x^{2} - 4 x + 4$ $(y + 5)^{2} = y^{2} + 10 y + 25$

Expand kyun karein? General form mein squared binomials nahi hote, sirf $x^{2}, y^{2}, x, y, c$ terms hote hain.

Step 2: Substitute karo aur collect karo: $x^{2} - 4 x + 4 + y^{2} + 10 y + 25 = 16$ $x^{2} + y^{2} - 4 x + 10 y + 29 = 16$ $x^{2} + y^{2} - 4 x + 10 y + 13 = 0$

16 kyun move karein? General form right side ko zero set karti hai.

Verify: $2 g = - 4 ⟹ g = - 2$ , $2 f = 10 ⟹ f = 5$ , $c = 13$ .

Centre: $(- g, - f) = (2, - 5)$ ✓
Radius: $4 + 25 - 13 = 16 = 4$ ✓

[!example] Example 3: Koi Real Circle Nahi (Imaginary Case)

Problem: Kya $x^{2} + y^{2} + 2 x + 4 y + 20 = 0$ ek real circle represent karta hai?

Solution:

Step 1: Parameters extract karo:

$2 g = 2 ⟹ g = 1$
$2 f = 4 ⟹ f = 2$
$c = 20$

Step 2: $r^{2} = g^{2} + f^{2} - c$ check karo: $r^{2} = 1 + 4 - 20 = - 15$

Yeh problem kyun hai? Radius squared negative hai → koi real radius exist nahi karta.

Answer: Yeh ek imaginary circle hai (radius purely imaginary hai). Koi real point ise satisfy nahi karta.

Physical intuition: Constant $c$ bahut bada hai; centre aur radius ka data real space mein "contradictory" hai.

[!example] Example 4: Point Circle (Degenerate Case)

Problem: $x^{2} + y^{2} - 6 x + 4 y + 13 = 0$ ko analyze karo.

Solution:

$g = - 3$ , $f = 2$ , $c = 13$
$r^{2} = 9 + 4 - 13 = 0 ⟹ r = 0$

Answer: Centre $(- (- 3), - 2) = (3, - 2)$ , radius $0$ . Yeh ek point circle hai — circle ek single point mein "collapse" ho gayi hai. Yeh limiting case hai jab ek circle shrink hoti hai.

[!mistake] Common Mistakes & Steel-Manning

Mistake 1: Negative Signs Bhool Jaana

Galat approach: $x^{2} + y^{2} + 6 x - 8 y + 5 = 0 ⟹$ centre hai $(6, - 8)$ .

Kyun sahi lagta hai: Tum $+ 6 x$ dekhte ho aur sochte ho "centre ka $x$ -coordinate 6 hai."

Fix: Formula $(- g, - f)$ hai, $(g, f)$ nahi. General form mein $+ 6 x$ standard form ke $- 2 h x$ se correspond karta hai, toh $h = - 3$ .

Sahi: $2 g = 6 ⟹ g = 3 ⟹ h = - g = - 3$ . Centre hai $(- 3, 4)$ .

Mistake 2: $c$ ko Seedha $r^{2}$ Maanna

Galat approach: $x^{2} + y^{2} + 4 x - 2 y + 1 = 0 ⟹ r^{2} = 1$ .

Kyun sahi lagta hai: Constant term radius se related lagta hai.

Fix: Constant $c$ mein centre ki position bhi encode hoti hai: $c = h^{2} + k^{2} - r^{2}$ . Tumhe $r^{2} = g^{2} + f^{2} - c$ compute karna hoga.

Sahi: $g = 2, f = - 1, c = 1 ⟹ r^{2} = 4 + 1 - 1 = 4 ⟹ r = 2$ .

Mistake 3: Validity Check Na Karna

Galat approach: $r = g^{2} + f^{2} - c$ blindly likh dena, chahe negative ho.

Kyun sahi lagta hai: Algorithms mein tum rush karte ho; verify karna bhool jaate ho.

Fix: Hamesha check karo $g^{2} + f^{2} - c \geq 0$ . Agar negative ho, toh likho "no real circle." Agar zero ho, toh "point circle."

[!recall]- Ek 12-Saal ke Bachche ko Explain Karo

Socho tum ek treasure (circle ka centre) chupaate ho kisi ko clues (equation) dekar. Circle describe karne ka fancy tarika hai "saare points jo 5 steps door hain treasure se jo (3, 4) par hai" — yahi hai $(x - 3)^{2} + (y - 4)^{2} = 25$ .

Lekin agar tum ise expand karo (jaise ek wrapped gift kholna), toh milti hai ek messy equation jaise $x^{2} + y^{2} - 6 x - 8 y + 0 = 0$ . Ab treasure ki location (3, 4) aur distance (5 steps) un numbers $- 6, - 8, 0$ mein scramble ho gayi hain.

Treasure dhundhne ke liye, tum complete the square karte ho — yeh jaise gift ko dobara wrap karna hai. Tum $x$ wali cheezein group karte ho: $x^{2} - 6 x$ ban jaata hai $(x - 3)^{2} - 9$ . $y$ ke liye bhi yehi. Phir dikhta hai: "Arre! Centre (3, 4) par hai aur radius 5 hai." $- 6$ secretly $- 2 \times 3$ tha, 3 ko chupaaye hua. General form sirf circle ka disguise hai!

[!mnemonic] Centre aur Radius Quick Recall

"Negative Guys Find Circles"

Negative: Centre mein coefficients ka negative hota hai
Guys: $g$ $x$ term se aata hai ( $2 g$ )
Find: $f$ $y$ term se aata hai ( $2 f$ )
Circles: Radius dhundhne ke liye $g^{2} + f^{2} - c$ use karo

Formula chant: "Centre is minus-g, minus-f; radius is root of g-squared, f-squared, minus c."

Diagram

Figure — General form of circle — converting, finding centre and radius

Connections

Standard form of circle — completing the square ke baad milne wali target form
Completing the square — conversion ke liye use ki jaane wali algebraic technique
Equation of circle from endpoints of diameter — aksar seedhi general form deta hai
Tangent to a circle — implicit differentiation se general form se nikalna aasaan hai
Family of circles — general form parameterized circles ke liye natural representation hai
Conic sections general equation — $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ ; circle special case hai jab $A = C, B = 0$

Flashcards

Circle equation ki general form kya hoti hai?

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

Circle

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

ka centre kya hai?

(- g, - f)

General form

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

ka radius formula kya hai?

r = g^{2} + f^{2} - c

General form mein real circle ke liye kaunsi condition hold karni chahiye?

g^{2} + f^{2} - c > 0

(agar zero ho toh point circle; agar negative ho toh imaginary)

x^{2} + y^{2} + 8 x - 6 y + 10 = 0

ka centre nikalo.

2 g = 8 ⟹ g = 4

;

2 f = - 6 ⟹ f = - 3

; centre hai

(- 4, 3)

x^{2} + y^{2} - 4 x + 10 y + 4 = 0

ka radius nikalo.

g = - 2, f = 5, c = 4

;

r = 4 + 25 - 4 = 25 = 5

Centre mein negative signs kyun hote hain:

(- g, - f)

Kyunki general form mein

2 g

, standard form

(x - h)^{2}

- 2 h

se correspond karta hai, toh

h = - g

(x + 1)^{2} + (y - 3)^{2} = 9

ko general form mein convert karo.

Expand karo:

x^{2} + 2 x + 1 + y^{2} - 6 y + 9 = 9 ⟹ x^{2} + y^{2} + 2 x - 6 y + 1 = 0

Point circle kya hota hai?

Jab

r = 0

(yaani

g^{2} + f^{2} = c

), circle ek single point mein degenerate ho jaati hai centre par

g^{2} + f^{2} - c < 0

ka geometrically kya matlab hai?

Equation ek imaginary circle represent karti hai jise satisfy karne wala koi real point nahi hota

General form of circle — converting, finding centre and radius

Overview

[!intuition] General Form Exist Kyun Karti Hai?

[!definition] General Form ke Components

[!formula] Conversion Formulas (Scratch se Derive Kiye Hue)

Derivation: Completing the Square

Final Result

[!example] Example 1: Standard Conversion

[!example] Example 2: Standard se General mein Convert Karna

[!example] Example 3: Koi Real Circle Nahi (Imaginary Case)

[!example] Example 4: Point Circle (Degenerate Case)

[!mistake] Common Mistakes & Steel-Manning

Mistake 1: Negative Signs Bhool Jaana

Mistake 2: $c$ ko Seedha $r^{2}$ Maanna

Mistake 3: Validity Check Na Karna

[!recall]- Ek 12-Saal ke Bachche ko Explain Karo

[!mnemonic] Centre aur Radius Quick Recall

Diagram

Connections

Flashcards

Concept Map

Deep Dive