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  • If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB isOpt

If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is

Option: 1

Parabola

 


Option: 2

Hyperbola 


Option: 3

Ellipse


Option: 4

Circle


Answers (1)

best_answer

Let P be the foot of perpendicular from O on AB, and let the center of the circle be C.

Since the circle passes through the origin, we have OC = R.

Let Q be the point of intersection of OC with AB. Then, OQ is perpendicular to AB.

Since OP is also perpendicular to AB, we have PQ || OA and PQ is perpendicular to OC.

Let x be the distance from A to P, and let y be the distance from P to B.

Then, by similar triangles, we have:

\begin{aligned} & x / R=R / y \\ x y= & R^2 \end{aligned}

Therefore, the locus of P is the hyperbola x y= R^2

To see this, note that as we move P along AB, the product xy remains constant and equal toR^2 This is the equation of a hyperbola with foci at(0,0)and(0,0) and y=xasymptotes and y=-x

Hence, the locus of the foot of perpendicular from O on AB is the hyperbolax y= R^2

Posted by

seema garhwal

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