Question

find the equation of the circle with
1. Centre (0, 2) and radius 2
2.Centre (1/2, 1/4) and radius (1/12)
​

Answer 1

Solution :-

Condition (I) :- Centre (0, 2) and radius 2.

↪Let us consider the equation of a circle with centre (h, k) and,

↪Radius r is given as (x - h)² + (y - k)² = r².

↪So, centre (h, k) = (0, 2) and radius (r) = 2.

↪The equation of the circle is :

↪(x - 0)² + (y - 2)² = 2²

↪x² + y² + 4 - 4y = 4

↪${\boxed{\red{\tt{x^2 + y^2 - 4y = 0}}}}$

∴ The equation of the circle is x² + y² -4y = 0.

_________________

Condition (2) :- Centre (1/2, 1/4) and radius (1/12).

↪Let us consider the equation of a circle with centre (h, k) and,

↪Radius r is given as (x – h)² + (y – k)² = r².

↪So, centre (h, k) = (1/2, 1/4) and radius (r) = 1/12.

↪The equation of the circle is :

↪(x – 1/2)² + (y – 1/4)² = (1/12)²

↪x² – x + ¼ + y² – y/2 + 1/16 = 1/144

↪x² - x + ¼ + y² – y/2 + 1/16 = 1/144

↪144x² - 144x + 36 + 144y² - 72y + 9 - 1 = 0

↪144x² - 144x + 144y² -72y + 44 = 0

↪36x² + 36x + 36y² - 18y + 11 = 0

↪${\boxed{\blue{\tt{36x^2 + 36y^2 - 36x - 18y + 11= 0}}}}$

∴ The equation of the circle is 36x² + 36y² - 36x - 18y + 11= 0.

Answer 2

sᴏʟᴜᴛɪᴏɴ :-

ᴄᴏɴᴅɪᴛɪᴏɴ (ɪ) :- ᴄᴇɴᴛʀᴇ (, ) ᴀɴᴅ ʀᴀᴅɪᴜs .

↪ʟᴇᴛ ᴜs ᴄᴏɴsɪᴅᴇʀ ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴ ᴏғ ᴀ ᴄɪʀᴄʟᴇ ᴡɪᴛʜ ᴄᴇɴᴛʀᴇ (ʜ, ᴋ) ᴀɴᴅ,

↪ʀᴀᴅɪᴜs ʀ ɪs ɢɪᴠᴇɴ ᴀs (x - ʜ)² + (ʏ - ᴋ)² = ʀ².

↪sᴏ, ᴄᴇɴᴛʀᴇ (ʜ, ᴋ) = (, ) ᴀɴᴅ ʀᴀᴅɪᴜs (ʀ) = .

↪ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴄɪʀᴄʟᴇ ɪs :

↪(x - )² + (ʏ - )² = ²

↪x² + ʏ² + - ʏ =

↪{\ʙᴏxᴇᴅ{\ʀᴇᴅ{\ᴛᴛ{x^ + ʏ^ - ʏ = }}}}

x

+ʏ

−ʏ=

∴ ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴄɪʀᴄʟᴇ ɪs x² + ʏ² -ʏ = .

_________________

ᴄᴏɴᴅɪᴛɪᴏɴ () :- ᴄᴇɴᴛʀᴇ (/, /) ᴀɴᴅ ʀᴀᴅɪᴜs (/).

↪ʟᴇᴛ ᴜs ᴄᴏɴsɪᴅᴇʀ ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴ ᴏғ ᴀ ᴄɪʀᴄʟᴇ ᴡɪᴛʜ ᴄᴇɴᴛʀᴇ (ʜ, ᴋ) ᴀɴᴅ,

↪ʀᴀᴅɪᴜs ʀ ɪs ɢɪᴠᴇɴ ᴀs (x – ʜ)² + (ʏ – ᴋ)² = ʀ².

↪sᴏ, ᴄᴇɴᴛʀᴇ (ʜ, ᴋ) = (/, /) ᴀɴᴅ ʀᴀᴅɪᴜs (ʀ) = /.

↪ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴄɪʀᴄʟᴇ ɪs :

↪(x – /)² + (ʏ – /)² = (/)²

↪x² – x + ¼ + ʏ² – ʏ/ + / = /

↪x² - x + ¼ + ʏ² – ʏ/ + / = /

↪x² - x + + ʏ² - ʏ + - =

↪x² - x + ʏ² -ʏ + =

↪x² + x + ʏ² - ʏ + =

↪{\ʙᴏxᴇᴅ{\ʙʟᴜᴇ{\ᴛᴛ{x^ + ʏ^ - x - ʏ + = }}}}

x

+ʏ

−x−ʏ+=

∴ ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴄɪʀᴄʟᴇ ɪs x² + ʏ² - x - ʏ + = .

Step-by-step explanation: