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(B) Let the foot of the perpendicular from the origin $O(0,0)$ to the line $AB$ be $P(h, k).$ Since $OP \perp AB,$ the slope of $OP$ is $m_1 = \frac{k

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$({x^2} + {y^2})^3 = 4{R^2}{x^2}{y^2}$

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(B) Let the foot of the perpendicular from the origin $O(0,0)$ to the line $AB$ be $P(h, k).$ Since $OP \perp AB,$ the slope of $OP$ is $m_1 = \frac{k}{h}.$ Thus,the slope of $AB$ is $m_2 = -\frac{h}{k}.$ The equation of line $AB$ passing through $P(h, k)$ is $y - k = -\frac{h}{k}(x - h),$ which simplifies to $hx + ky = h^2 + k^2.$ The intercepts of this line on the axes are $A\left(\frac{h^2 + k^2}{h}, 0\right)$ and $B\left(0, \frac{h^2 + k^2}{k}\right).$ Since $AB$ is a chord of the circle with radius $R$ and the angle $\angle AOB = 90^\circ,$ $AB$ is a diameter of the circle. Thus,the length $AB = 2R.$ Using the distance formula,$AB^2 = (2R)^2 = 4R^2.$ $\left(\frac{h^2 + k^2}{h}\right)^2 + \left(\frac{h^2 + k^2}{k}\right)^2 = 4R^2.$ $(h^2 + k^2)^2 \left(\frac{1}{h^2} + \frac{1}{k^2}\right) = 4R^2.$ $(h^2 + k^2)^2 \left(\frac{h^2 + k^2}{h^2k^2}\right) = 4R^2.$ $(h^2 + k^2)^3 = 4R^2h^2k^2.$ Replacing $(h, k)$ with $(x, y),$ the locus is $(x^2 + y^2)^3 = 4R^2x^2y^2.$

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 the perpendicular from $O$ on $AB$ is

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