Let $A_1, A_2, \ldots, A_n$ be distinct points in the plane. Suppose that each point $A_i$ can be assigned a real number $\lambda_i \neq 0$ in such a way that

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Let $A_1, A_2, \ldots, A_n$ be distinct points in the plane. Suppose that each point $A_i$ can be assigned a real number $\lambda_i \neq 0$ in such a way that
\[
A_i A_j^2=\lambda_i+\lambda_j, \quad \text { for all } i, j \text { with } i \neq j .
\]

(a) Show that $n \leq 4$.

(b) Prove that if $n=4$, then $\frac{1}{\lambda_1}+\frac{1}{\lambda_2}+\frac{1}{\lambda_3}+\frac{1}{\lambda_4}=0$.

asked
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Sep 23 in Mathematics by ♦MathsGee Platinum (138,598 points) | 21 views

Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /home/customer/www/mathsgee.com/public_html/qa-include/qa-theme-base.php on line 935

Deprecated: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in /home/customer/www/mathsgee.com/public_html/qa-include/qa-theme-base.php on line 935

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Let \(A_1, A_2, \ldots, A_n\) be distinct points in the plane. Suppose that each point \(A_i\) can be assigned a real number \(\lambda_i \neq 0\) in such a way that

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Let \(A_1, A_2, \ldots, A_n\) be distinct points in the plane. Suppose that each point \(A_i\) can be assigned a real number \(\lambda_i \neq 0\) in such a way that

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