Engineering Entrance IOQM Coordinate Geometry and others

𝑇ℎ𝑒 𝑠ℎ𝑜𝑒𝑙𝑎𝑐𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎, 𝑎𝑙𝑠𝑜 𝑘𝑛𝑜𝑤𝑛 𝑎𝑠 𝐺𝑎𝑢𝑠𝑠′𝑠 𝑎𝑟𝑒𝑎 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑢𝑟𝑣𝑒𝑦𝑜𝑟′𝑠 𝑓𝑜𝑟𝑚𝑢𝑙𝑎.
𝐼𝑓 𝑥 (−1, −2), 𝑦 (6, 1), 𝑧 (3, 5) 𝑎𝑟𝑒 𝑚𝑖𝑑 𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝐴𝐵𝐶, 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑖𝑠
𝑇ℎ𝑒 𝑚𝑖𝑑−𝑝𝑜𝑖𝑛𝑡𝑠 𝐷, 𝐸, 𝐹 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑑𝑒𝑠 𝑜𝑓 𝑎 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝐴𝐵𝐶 𝑎𝑟𝑒 (3, 4), (8, 9)𝑎𝑛𝑑 (6, 7). 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒.
𝑇ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡, 𝑖𝑓 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑒𝑛𝑡𝑎𝑔𝑜𝑛 𝑃𝑄𝑅𝑆𝑇 𝑏𝑒 30 𝑆𝑞. 𝑈𝑛𝑖𝑡𝑠 𝑤ℎ𝑒𝑟𝑒 𝑃=(2, 3), 𝑄=(−3, 1) R=(−2, 4), S=(0, 1) and T=(3, t) are ___
𝑇ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ (1,2)(3,−4)(5,−6) 𝑖𝑠
[𝐵𝐼𝑇𝑆𝐴𝑇] 𝑇ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒𝑠 3𝑥 + 4𝑦 + 7 = 0 𝑎𝑛𝑑 12𝑥 + 5𝑦−8 = 0 𝑎𝑟𝑒 :
[𝐵𝐼𝑇𝑆𝐴𝑇2020] 𝑇ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 sin^4⁡〖𝑥−(𝑘+2) sin^2⁡〖𝑥−(𝑘+3)=0 𝑝𝑜𝑠𝑠𝑒𝑠𝑠 𝑎 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑓, 〗 〗
𝐴. 𝒌 greater than−𝟑 𝐵. 𝒌 less than −𝟐 𝐶. −𝟑≤𝒌≤−𝟐 𝐷. 𝒌 𝒊𝒔+𝒗𝒆
[𝐵𝐼𝑇𝑆𝐴𝑇2021] 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒𝑠 𝑜𝑓 𝑟𝑜𝑜𝑡𝑠 𝑜𝑓
𝑥^2−(𝑎−2)𝑥−(𝑎+1)=0 𝑖𝑠 𝑙𝑒𝑎𝑠𝑡.
[𝐵𝐼𝑇𝑆𝐴𝑇2023] 𝐼𝑓 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝐺𝑃 𝑎, 𝑎𝑟, 𝑎𝑟^2…. 𝑖𝑠 15, 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑠𝑞𝑢𝑎𝑟𝑒𝑠 𝑜𝑓 𝑖𝑡𝑠 𝑒𝑎𝑐ℎ 𝑡𝑒𝑟𝑚 𝑖𝑠 150,
𝒕𝒉𝒆𝒏 𝒕𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝒂𝒓^𝟐, 𝒂𝒓^𝟒, 𝒂𝒓^𝟔, …. 𝒊𝒔 ?