Question

the centre of the circle which passes through the focus of the parabola X^2=4y and touches it at the point (6,9) is

Answer 1

Let centre of circle is (h, r) And radius of circle is r
∴equation of circle is (x - h)² + (y - r)² =

A/C to question,
Circle passing through focus and touches at the point (6,9) of parabola x² = 4y
focus of parabola is (0,1)
so, (0,1) will satisfy the equation of circle ,
∴ (0 - h)² + (1 - k)² = r² -----(1)
Again, (6, 9) will satisfy the equation of circle ,
∴(6 - h)² + (9 - k)² = r² ------(2)

From equations (1) and (2),
-2k + 1 = -12h + 36 - 18k + 81
⇒16k + 12h = 116
⇒4h + 3h = 29 ------(3)


at (6,9) slope of tangent if circle = slope of tangent of parabola [because at (6,9) circle touches the parabola ]
slope of circle :
differentiate equation of circle with respect to x
2(x - h) + 2(y - k)dy/dx = 0
⇒ dy/dx = -(x - h)/(y - k)
At (6,9) slope of tangent of circle is dy/dx = -(6 -h)/(9 - k)

Slope of parabola :
x² = 4y, differentiate with respect to x
2x = 4dy/dx
⇒dy/dx = x/2
At (6,9) slope of tangent of parabola is dy/dx = 6/2 = 3

Hence, 3 = -(6 - h)/(9 - k)
⇒3(9 - k) + (6 - h) = 0
⇒ 27 - 3k + 6 - h = 0
⇒ 3k + h = 33 -------(4)

Solve equation (3) and (4) ,
h = -9 and k = 14
so, r² = (6 +9)² + (9 - 14)² = 225 + 25 = 250

Hence, equation of circle is (x + 9)² + (y - 14)² = 250
⇒x² + y² + 18x - 28y + 81 + 196 = 250
⇒x² + y² + 18x - 28y + 27 = 0

Hence, answer is x² + y² + 18x - 28y + 27 = 0

Answer 2

▆ ▇ █ ⓈⓄⓁⓊⓉⒾⓄⓃ █ ▇ ▆

ʟᴇᴛ ᴄᴇɴᴛʀᴇ ᴏꜰ ᴄɪʀᴄʟᴇ ɪꜱ (ʜ, ʀ) ᴀɴᴅ ʀᴀᴅɪᴜꜱ ᴏꜰ ᴄɪʀᴄʟᴇ ɪꜱ ʀ

∴ᴇQᴜᴀᴛɪᴏɴ ᴏꜰ ᴄɪʀᴄʟᴇ ɪꜱ (x - ʜ)² + (ʏ - ʀ)² =

ᴀ/ᴄ ᴛᴏ Qᴜᴇꜱᴛɪᴏɴ,

ᴄɪʀᴄʟᴇ ᴘᴀꜱꜱɪɴɢ ᴛʜʀᴏᴜɢʜ ꜰᴏᴄᴜꜱ ᴀɴᴅ ᴛᴏᴜᴄʜᴇꜱ ᴀᴛ ᴛʜᴇ ᴘᴏɪɴᴛ (6,9) ᴏꜰ ᴘᴀʀᴀʙᴏʟᴀ x² = 4ʏ

ꜰᴏᴄᴜꜱ ᴏꜰ ᴘᴀʀᴀʙᴏʟᴀ ɪꜱ (0,1)

ꜱᴏ, (0,1) ᴡɪʟʟ ꜱᴀᴛɪꜱꜰʏ ᴛʜᴇ ᴇQᴜᴀᴛɪᴏɴ ᴏꜰ ᴄɪʀᴄʟᴇ ,

∴ (0 - ʜ)² + (1 - ᴋ)² = ʀ² -----(1)

ᴀɢᴀɪɴ, (6, 9) ᴡɪʟʟ ꜱᴀᴛɪꜱꜰʏ ᴛʜᴇ ᴇQᴜᴀᴛɪᴏɴ ᴏꜰ ᴄɪʀᴄʟᴇ ,

∴(6 - ʜ)² + (9 - ᴋ)² = ʀ² ------(2)

ꜰʀᴏᴍ ᴇQᴜᴀᴛɪᴏɴꜱ (1) ᴀɴᴅ (2),

-2ᴋ + 1 = -12ʜ + 36 - 18ᴋ + 81

⇒16ᴋ + 12ʜ = 116

⇒4ʜ + 3ʜ = 29 ------(3)

ᴀᴛ (6,9) ꜱʟᴏᴘᴇ ᴏꜰ ᴛᴀɴɢᴇɴᴛ ɪꜰ ᴄɪʀᴄʟᴇ = ꜱʟᴏᴘᴇ ᴏꜰ ᴛᴀɴɢᴇɴᴛ ᴏꜰ ᴘᴀʀᴀʙᴏʟᴀ [ʙᴇᴄᴀᴜꜱᴇ ᴀᴛ (6,9) ᴄɪʀᴄʟᴇ ᴛᴏᴜᴄʜᴇꜱ ᴛʜᴇ ᴘᴀʀᴀʙᴏʟᴀ ]

ꜱʟᴏᴘᴇ ᴏꜰ ᴄɪʀᴄʟᴇ :

ᴅɪꜰꜰᴇʀᴇɴᴛɪᴀᴛᴇ ᴇQᴜᴀᴛɪᴏɴ ᴏꜰ ᴄɪʀᴄʟᴇ ᴡɪᴛʜ ʀᴇꜱᴘᴇᴄᴛ ᴛᴏ x

2(x - ʜ) + 2(ʏ - ᴋ)ᴅʏ/ᴅx = 0

⇒ ᴅʏ/ᴅx = -(x - ʜ)/(ʏ - ᴋ)

ᴀᴛ (6,9) ꜱʟᴏᴘᴇ ᴏꜰ ᴛᴀɴɢᴇɴᴛ ᴏꜰ ᴄɪʀᴄʟᴇ ɪꜱ ᴅʏ/ᴅx = -(6 -ʜ)/(9 - ᴋ)

ꜱʟᴏᴘᴇ ᴏꜰ ᴘᴀʀᴀʙᴏʟᴀ :

x² = 4ʏ, ᴅɪꜰꜰᴇʀᴇɴᴛɪᴀᴛᴇ ᴡɪᴛʜ ʀᴇꜱᴘᴇᴄᴛ ᴛᴏ x

2x = 4ᴅʏ/ᴅx

⇒ᴅʏ/ᴅx = x/2

ᴀᴛ (6,9) ꜱʟᴏᴘᴇ ᴏꜰ ᴛᴀɴɢᴇɴᴛ ᴏꜰ ᴘᴀʀᴀʙᴏʟᴀ ɪꜱ ᴅʏ/ᴅx = 6/2 = 3

ʜᴇɴᴄᴇ, 3 = -(6 - ʜ)/(9 - ᴋ)

⇒3(9 - ᴋ) + (6 - ʜ) = 0

⇒ 27 - 3ᴋ + 6 - ʜ = 0

⇒ 3ᴋ + ʜ = 33 -------(4)

ꜱᴏʟᴠᴇ ᴇQᴜᴀᴛɪᴏɴ (3) ᴀɴᴅ (4) ,

ʜ = -9 ᴀɴᴅ ᴋ = 14

ꜱᴏ, ʀ² = (6 +9)² + (9 - 14)² = 225 + 25 = 250

ʜᴇɴᴄᴇ, ᴇQᴜᴀᴛɪᴏɴ ᴏꜰ ᴄɪʀᴄʟᴇ ɪꜱ (x + 9)² + (ʏ - 14)² = 250

⇒x² + ʏ² + 18x - 28ʏ + 81 + 196 = 250

⇒x² + ʏ² + 18x - 28ʏ + 27 = 0

ʜᴇɴᴄᴇ, ᴀɴꜱᴡᴇʀ ɪꜱ x² + ʏ² + 18x - 28ʏ + 27 = 0