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Question

Find the value of $$\dfrac { { a }^{ 2 }-{ b }^{ 2 }-{ c }^{ 2 } }{ \left( a-b \right) \left( a-c \right)  } +\dfrac { { b }^{ 2 }-{ c }^{ 2 }-{ a }^{ 2 } }{ \left( b-c \right) \left( b-a \right)  } +\dfrac { { c }^{ 2 }-{ a }^{ 2 }-{ b }^{ 2 } }{ \left( c-a \right) \left( c-b \right)  } $$.


Solution

$$\dfrac{a^{2}-b^{2}-c^{2}}{(a-b)(a-c)}$$$$+$$$$\dfrac{b^{2}-c^{2}-a^{2}}{(b-c)(b-a)}$$$$+$$ $$\dfrac{c^{2}-a^{2}-b^{2}}{(c-a)(c-b)}$$

$$=$$$$\dfrac{a^{2}-b^{2}-c^{2}}{(a-b)(a-c)}$$ $$-$$ $$\dfrac{b^{2}-c^{2}-a^{2}}{(b-c)(a-b)}$$ $$+$$ $$\dfrac{c^{2}-a^{2}-b^{2}}{(c-a)(c-b)}$$


$$=$$$$\dfrac{(a^{2}-b^{2}-c^{2})(b-c)-(b^{2}-c^{2}-a^{2})(a-c)+(c^{2}-a^{2}-b^{2})(a-b)}{(a-b)(a-c)(b-c)}$$
Expanding numerator and denominator we get,

$$=$$ $$\dfrac{a^{2}b-a^{2}c-b^{3}+b^{2}c-c^{2}b+c^{3}-ab^{2}+b^{2}c+ac^{2}-c^{3}+a^{3}-a^{2}c+ac^{2}-bc^{2}-a^{3}+a^{2}b+b^{3}-ab^{2}}{a^{2}b-a^{2}c-abc+ac^{2}-ab^{2}+abc+b^{2}c-bc^{2}}$$


$$=$$ $$\dfrac{2(a^{2}b-a^{2}c+ac^{2}-ab^{2}+b^{2}c-bc^{2})}{a^{2}b-a^{2}c+ac^{2}-ab^{2}+b^{2}c-bc^{2}}$$

$$=2$$

Mathematics

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