# Answer to Question #16831 in Calculus for soni sharma

Question #16831

A series of circles have their centres on a given straight line and their radii proportional tothe distances

of their corresponding centres from a given point in that line .find the envelope.plz solve this question step by step.

of their corresponding centres from a given point in that line .find the envelope.plz solve this question step by step.

Expert's answer

Let the given point is A. Let O is a center of typicalcircle. AC is the envelope.

The radius of circle OC = AO * k.

The angle ACO = 90 degrees.

So, AC^2 + OC^2 = AO^2.

AC = sqrt(AO^2 - CO^2) = sqrt(AO^2 - AO^2 * k^2) = AO *sqrt(1 - k^2).

The radius of circle OC = AO * k.

The angle ACO = 90 degrees.

So, AC^2 + OC^2 = AO^2.

AC = sqrt(AO^2 - CO^2) = sqrt(AO^2 - AO^2 * k^2) = AO *sqrt(1 - k^2).

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