Question:১২.>(a + b + c)p = (b + c - a)q = (c + a - b)r = (a + b - c)s হলে----- ক. ধ্রবক k এর মাধ্যমে `1/p` এর মান নির্ণয় কর। খ. প্রমাণ কর যে, `1/q + 1/r + 1/s = 1/p` গ. প্রমাণ কর যে, `(1/p^2 - 1/s^2) + (1q^2 - r^2) = (8bc)/k^2`
Answer ক. ধরি, (a + b + c)p = (b + c - a)q = (c + a - b)r = (a + b - c)s = k :. (a + b + c)p = k বা, `(a + b + c)/k = 1/p` :.` 1/p = (a + b + c)/k` খ. ‘ক’ হতে পাই, :. `1/p = (a + b + c)/k`..........(i) আবার q (b + c - a) = k q = `k/(b + c - a)` :.` 1/q = (b + c - a)/k`..............(ii) এবং r (c + a - b) = k `r = k/(c + a - b)` :. `1/r = (c + a - b)/k`..............(iii) আবার s(a + b - c) = k `s = k/(a + b - c)` :.` 1/s = (a + b - c)/k`................(iv) (ii) নং (iii) নং এবং (iv) সমীকরণ যোগ করে পাই, `1/q + 1/r + 1/s = (b + c - a)/k + (c + a - b)/k + (a + b - c)/k` =`1/k (b + c - a + c + a - b + a + b - c)` =` (a + b + c)/k` =` 1/p` [(i) নং থেকে] :.` 1/q + 1/r + 1/s = 1/p` (প্রমাণিত) গ. ‘ক’ ও ‘খ’ হতে পাই, `1/p = (a + b + c)/k` `1/q = (b + c - a)/k` `1/r = (c + a - b)/k` এবং `1/s = (a + b - c)/k` এখন, `1/p^2 - 1/s^2 = ((a + b + c)/k)^2 - ((a + b - c)/k)^2` `= (a^2 + b^2 + c^2 + 2ab + 2bc + 2ca)/(k^2) - (a^2 + b^2 + c^2 + 2ab - 2bc - 2ca)/k^2` `= (a^2 + b^2 + c^2 + 2ab + 2bc + 2ca - a^2 - b^2 - c^2 - 2ab + 2bc + 2ca)/k^2` `= (4bc + 4ca)/k^2` আবার, `1/q^2 - 1/r^2 = ((b + c - a)/k)^2 - ((c + a - b)/k)^2` `= (b^2 + c^2 + a^2 + 2bc - 2ca - 2ab)/(k^2) - (c^2 + a^2 + b^2 + 2ca - 2ab - 2bc)/k^2` `= (b^2 + c^2 + a^2 + 2bc - 2ca - 2ab - c^2 - a^2 - b^2 - 2ca + 2ab + 2bc)/k^2` `= (4bc - 4ca)/k^2` :.` (1/p^2 - 1/s^2) + (1/q^2 - 1/r^2) = (4bc + 4ca)/k^2 + (4bc - 4ca)/k^2` =` (4bc + 4ca + 4bc - 4ca)/k^2 = (8bc)/k^2` :. `(1/p^2 - 1/s^2) + (1/q^2 - 1/r^2) = (8bc)/k^2` (প্রমাণিত)
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১২.>(a + b + c)p = (b + c - a)q = (c + a - b)r = (a + b - c)s hole----- ka. dhrbok k ar madhjome `1/p` ar man nirony karo. kh. proman kar je, `1/q + 1/r + 1/s = 1/p` ga. proman kar je, `(1/p^2 - 1/s^2) + (1q^2 - r^2) = (8bc)/k^2`