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The locus of a point \(P\) such that \(P A+P B=4\) where \(A(2,3,4), B(-2,3,4)\) is
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1742 Upvotes
Verified Answer
The correct answer is:
\(y^2+z^2-6 y-8 z+25=0\)
Given points are \(A(2,3,4)\) and \(B(-2,3,4)\) and \(A B=4\).
Now let point \(P(x, y, z)\), such that \(P A+P B=4=A B\) means point \(P\) is collinear with points \(A\) and \(B\) and lies between them, so
\(\begin{array}{ll}
& \frac{x-2}{2+2}=\frac{y-3}{3-3}=\frac{z-4}{4-4} \\
\Rightarrow & y-3=0=z-4 \\
\text {or } & (y-3)^2+(z-4)^2=0 \\
\Rightarrow & y^2+z^2-6 y-8 z+25=0
\end{array}\)
Hence, option (c) is correct.
Now let point \(P(x, y, z)\), such that \(P A+P B=4=A B\) means point \(P\) is collinear with points \(A\) and \(B\) and lies between them, so
\(\begin{array}{ll}
& \frac{x-2}{2+2}=\frac{y-3}{3-3}=\frac{z-4}{4-4} \\
\Rightarrow & y-3=0=z-4 \\
\text {or } & (y-3)^2+(z-4)^2=0 \\
\Rightarrow & y^2+z^2-6 y-8 z+25=0
\end{array}\)
Hence, option (c) is correct.
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