A line segment joining (1, 0, 1) and the origin (0, 0, 0) is revolved about the x-axis to form a right circular cone. If (x, y, z) is any point on the cone other than the origin, then it satisfies the equation

$x^{2} - 2 y^{2} - z^{2} = 0$

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$x^{2} - y^{2} - z^{2} = 0$

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$2 x^{2} - y^{2} - 2 z^{2} = 0$

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$x^{2} - 2 y^{2} - 2 z^{2} = 0$

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Solution

The correct option is B
$x^{2} - y^{2} - z^{2} = 0$

Let A be the point (1,0,1) and P(x, y, z) be any point on the cone obtained revolving OA about the x-axis.
From the figure,
$Δ O P M a n d Δ O A C a r e s i m i l a r \frac{O P}{O A} = \frac{O M}{O C} \frac{\sqrt{x^{2} + y^{2} + z^{2}}}{\sqrt{2}} = \frac{x}{1} \Rightarrow x^{2} + y^{2} + z^{2} = 2 x^{2} \Rightarrow x^{2} - y^{2} - z^{2} = 0$

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A line segment joining (1, 0, 1) and the origin (0, 0, 0) is revolved about the x-axis to form a right circular cone. If (x, y, z) is any point on the cone other than the origin, then it satisfies the equation

The correct option is B x2−y2−z2=0 Let A be the point (1,0,1) and P(x, y, z) be any point on the cone obtained revolving OA about the x-axis. From the figure, ΔOPM and ΔOAC are similarOPOA=OMOC√x2+y2+z2√2=x1⇒x2+y2+z2=2x2⇒x2−y2−z2=0