If x- y are two non-zero and non-collinear vectors satisfying -a-2-2-b-3-c-x-a-2-2-b-3-c-y-a-2-2-b-3-c-x-y-0- where - - - are three distinct real numbers- then which of the following statements is-are correct -

Given : nbsp; [(a 2)α^2+(b 3)α+c]x+[(a 2)β+(b 3)β+c]y+[(a 2)γ^2+(b 3)γ+c](x×y)=0, Since x,y are non collinear vectors, ⇒x,y and x×y are non coplanar vectors. ...

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Standard XII

Mathematics

Condition for Two Lines to be on the Same Plane

If x, y are t...

Question

If $\to x, \to y$ are two non-zero and non-collinear vectors satisfying $[(a - 2) α^{2} + (b - 3) α + c] \to x + [(a - 2) β^{2} + (b - 3) β + c] \to y + [(a - 2) γ^{2} + (b - 3) γ + c] (\to x \times \to y) = \to 0,$ where $α, β, γ$ are three distinct real numbers, then which of the following statement(s) is/are correct ?

a (b + c - a) = 2

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b (c + a - b) = 3

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c (a + b - c) = 0

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a^{2} + b^{2} - c^{2} = 13

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Solution

The correct option is D $a^{2} + b^{2} - c^{2} = 13$
Given :
$[(a - 2) α^{2} + (b - 3) α + c] \to x + [(a - 2) β + (b - 3) β + c] \to y + [(a - 2) γ^{2} + (b - 3) γ + c] (\to x \times \to y) = \to 0,$
Since $\to x, \to y$ are non-collinear vectors,
$\Rightarrow \to x, \to y$ and $\to x \times \to y$ are non-coplanar vectors.
So, cofficient of each vectors will be zero.
$(a - 2) α^{2} + (b - 3) α + c = 0 (a - 2) β^{2} + (b - 3) β + c = 0 (a - 2) γ^{2} + (b - 3) γ + c = 0$
So, $α, β, γ$ are three distnict real roots of the equation $(a - 2) x^{2} + (b - 3) x + c = 0$
A quadratic equation with more than 2 roots will be an identity.
$\Rightarrow a = 2, b = 3, c = 0$

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Condition for Two Lines to be on the Same Plane

Standard XII Mathematics

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If →x,→y are two non-zero and non-collinear vectors satisfying [(a−2)α2+(b−3)α+c]→x+[(a−2)β2+(b−3)β+c]→y+[(a−2)γ2+(b−3)γ+c](→x×→y)=→0, where α,β,γ are three distinct real numbers, then which of the following statement(s) is/are correct ?