Find the perpendicular orbits of the family of circles whose center is on the line y=x and passes through its origin.

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Capodarcod0f · Accepted Answer

Given: the family of circles whose center is on the line y=x and passes through its origin.To find: The differential equation of the family of circles whose center is on the line y=x and passes through its origin.Concept Used: The equation of the circle having center as (h,k) and radius a is (x−h)2+(y−k)2=a2, using this and substituting appropiate the value of h,k and solving accordingly.ExplanationLet the center of the circle is (h,k) and radius is a, as per the question,circles whose center is on the line y=x and passes through its origin.So, we can write as h=k and a2=h2+k2.Now, from the equation of circle can be written as,(x−h)2+(y−h)2=h2+k2x2+h2−2xh+y2+h2−2yh=h2+h2x2+y2−2xh−2yh=0Now, differentiate the above expression w.r.t. x, we get,2x+2y⋅dydx−2⋅1⋅h−2⋅h⋅dydx=0dydx(2y−2h)−2h+2x=0Answer: The differential equation of the family of circles whose center is on the line y=x and passes through its origin is dydx(2y−2h)−2h+2x=0.

Find the perpendicular orbits of the family of circles whose

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