Question

Xy and x'y' are two parallel tangents to a centre with centre o and another tangent ab with point of contact c intersect xy at a and x'y' at
b. Prove that angleaob =90`

Answer 1

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Answer 2

Answer:

• Qᴜᴇꜱᴛɪᴏɴ:

ɪɴ ꜰɪɢ., xʏ ᴀɴᴅ x'ʏ' ᴀʀᴇ ᴛᴡᴏ ᴘᴀʀᴀʟʟᴇʟ ᴛᴀɴɢᴇɴᴛꜱ ᴛᴏ ᴀ ᴄɪʀᴄʟᴇ ᴡɪᴛʜ ᴄᴇɴᴛʀᴇ ᴏ ᴀɴᴅ ᴀɴᴏᴛʜᴇʀ ᴛᴀɴɢᴇɴᴛ ᴀʙ ᴛᴏᴜᴄʜɪɴɢ ᴀᴛ ᴄ, ɪɴᴛᴇʀꜱᴇᴄᴛɪɴɢ xʏ ᴀᴛ ᴀ ᴀɴᴅ x'ʏ' ᴀᴛ ʙ. ᴘʀᴏᴠᴇ ᴛʜᴀᴛ ᴢᴀᴏʙ = 90°.

• ꜱᴏʟᴜᴛɪᴏɴ :

ʟᴇᴛ ᴜꜱ ᴊᴏɪɴ ᴘᴏɪɴᴛ ᴏ ᴛᴏ ᴄ.

ɪɴ ᴀᴏᴘᴀ ᴀɴᴅ ᴀᴏᴄᴀ,

ᴏᴘ = ᴏᴄ (ʀᴀᴅɪɪ ᴏꜰ ᴛʜᴇ ꜱᴀᴍᴇ ᴄɪʀᴄʟᴇ)

ᴀᴘ = ᴀᴄ (ᴛᴀɴɢᴇɴᴛꜱ ꜰʀᴏᴍ ᴘᴏɪɴᴛ ᴀ)

ᴀᴏ = ᴀᴏ (ᴄᴏᴍᴍᴏɴ ꜱɪᴅᴇ)

ᴀᴏᴘᴀ = ᴀᴏᴄᴀ (ꜱꜱꜱ ᴄᴏɴɢʀᴜᴇɴᴄᴇ ᴄʀɪᴛᴇʀɪᴏɴ)

ᴛʜᴇʀᴇꜰᴏʀᴇ, ᴘ ᴄ, ᴀ + ᴀ, ᴏ - ᴏ

ᴢᴘᴏᴀ = ᴢᴄᴏᴀ ... (1)

ꜱɪᴍɪʟᴀʀʟʏ, ᴀᴏQʙ = ᴀᴏᴄʙ

ᴢQᴏʙ = 2ᴄᴏʙ . (ɪɪ) ...

ꜱɪɴᴄᴇ ᴘᴏQ ɪꜱ ᴀ ᴅɪᴀᴍᴇᴛᴇʀ ᴏꜰ ᴛʜᴇ ᴄɪʀᴄʟᴇ, ɪᴛ ɪꜱ ᴀ ꜱᴛʀᴀɪɢʜᴛ ʟɪɴᴇ.

ᴛʜᴇʀᴇꜰᴏʀᴇ, ᴘᴏᴀ + 2ᴄᴏᴀ + ᴄᴏʙ + ᴢQᴏʙ = 180°

ꜰʀᴏᴍ ᴇQᴜᴀᴛɪᴏɴꜱ (ɪ) ᴀɴᴅ (ɪɪ), ɪᴛ ᴄᴀɴ ʙᴇ ᴏʙꜱᴇʀᴠᴇᴅ ᴛʜᴀᴛ

2∠ᴄᴏᴀ+2∠ᴄᴏʙ=180∘

∠ᴄᴏᴀ+∠ᴄᴏʙ=90∘

∠ᴀᴏʙ=90∘