If x→,y→ are two $$non-zero$$ and $$non-collinear$$ vectors satisfying (a$$−$$2)α$$2+(b$$−$$3)$$α$$+cx$$→$$+(a$$−$$2)$$β$$2+(b$$−$$3)$$β$$+cy$$→$$+(a$$−$$2)$$γ$$2+(b$$−$$3)$$γ$$+c(x$$→×y→$$)=0$$ where α,β,γ are three distinct real numbers, then find the value of $$a2+b2+c2$$−$$4$$

Question

If x→,y→  are two $$non-zero$$ and $$non-collinear$$ vectors satisfying  (a$$−$$2)α$$2+(b$$−$$3)$$α$$+cx$$→$$+(a$$−$$2)$$β$$2+(b$$−$$3)$$β$$+cy$$→$$+(a$$−$$2)$$γ$$2+(b$$−$$3)$$γ$$+c(x$$→×y→$$)=0$$ where α,β,γ are three distinct real numbers, then find the value of $$a2+b2+c2$$−$$4$$

Infinity Learn · Accepted Answer

Since x→ and y→ are $$non-collinear$$ vectors, therefore x→,y→ and x→×y→ are $$non-coplanar$$ vectors. (a$$−$$2)α$$2+(b$$−$$3)$$α$$+c+(a$$−$$2)$$β$$2+(b$$−$$3)$$ββ$$+c$$]y→$$+(a$$−$$2)$$γ$$2+(b$$−$$3)$$γ$$+c(x$$→×y→$$)=0$$ Coefficient of each vector x→,y→ and x→×y→ is zero. (a$$−$$2)α$$2+(b$$−$$3)$$α$$+c=0(a$$−$$2)$$β$$2+(b$$−$$3)$$β$$+c=0(a$$−$$2)$$γ$$2+(b$$−$$3)$$β$$+c=0The$$ above three equations will satisfy if the coefficients of α,β and  γ are zero because α,β and γ are three distinct real numbers .a−$$2=0$$ or $$a=2b$$−$$3=0$$ or $$b=3$$ and $$c=0$$∴ $$a2+b2+c2=22+32+02=4+9=13$$

If x→,y→ are two non-zero and non-collinear vectors satisfying (a−2)α2+(b−3)α+cx→+(a−2)β2+(b−3)β+cy→+(a−2)γ2+(b−3)γ+c(x→×y→)=0 where α,β,γ are three distinct real numbers, then find the value of a2+b2+c2−4

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