Let a,b,c be distinct non-negative numbers and the vectors ai^+aj^+ck^,i^+k^,ci^+cj^+bk^ lie in a plane, and then prove that the quadratic equation ax2+2cx+b=0 has equal roots.

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Let {a},{b},{c} be distinct non-negative numbers and the vecto

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Let $a, b, c$ be distinct non-negative numbers and the vectors $a \hat{i} + a \hat{j} + c \hat{k}, \hat{i} + \hat{k}, c \hat{i} + c \hat{j} + b \hat{k}$ lie in a plane, and then prove that the quadratic equation $a x^{2} + 2 c x + b = 0$ has equal roots.

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Lucas Discussed

Let

a, b, c

be distinct non-negative numbers and the vectors

a \hat{i} + a \hat{j} + c \hat{k}, \hat{i} + \hat{k}, c \hat{i} + c \hat{j} + b \hat{k}

lie in a plane, and then prove that the quadratic equation

a x^{2} + 2 c x + b = 0

has equal roots.

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Question Text

Let

a, b, c

be distinct non-negative numbers and the vectors

a \hat{i} + a \hat{j} + c \hat{k}, \hat{i} + \hat{k}, c \hat{i} + c \hat{j} + b \hat{k}

lie in a plane, and then prove that the quadratic equation

a x^{2} + 2 c x + b = 0

has equal roots.