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Exercise 6.2 · Q25

Q.Find the differential equation of all circles having radius 99 and centre at point A(h,k)A(h,k).

Maharashtra MsbshseTextbookSubjectiveImportance★★★★★
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Circles of fixed radius 99 centred at (h,k)(h,k): (x−h)2+(y−k)2=81(x-h)^2+(y-k)^2=81, with h,kh,k arbitrary. Differentiate: (x−h)+(y−k)dydx=0⇒x−h=−(y−k)dydx(x-h)+(y-k)\dfrac{dy}{dx}=0\Rightarrow x-h=-(y-k)\dfrac{dy}{dx}. Differentiate again: 1+(dydx)2+(y−k)d2ydx2=0⇒(y−k)=−1+(dy/dx)2d2y/dx21+\left(\dfrac{dy}{dx}\right)^2+(y-k)\dfrac{d^2y}{dx^2}=0\Rightarrow (y-k)=-\dfrac{1+(dy/dx)^2}{d^2y/dx^2}. Substituting both into the circle equation and simplifying gives $\left[1+\left(\ …

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