If $(x-a)^2 + (y-b)^2 = c^2$ for some $c > 0$, then prove that $\dfrac{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}}{\dfrac{d^2y}{dx^2}}$ is independent of $a$ and $b$.

> > [!TLDR] > Differentiate the circle equation twice to eliminate $a$ and $b$, then substitute back into the original equation to isolate the given expression as a constant. > > Differentiating $(x-a)^2+(y-b)^2=c^2$ w.r.t. $x$: $(x-a)+(y-b)y_1=0$ ... (1) > …

If (x-a)² + (y-b)² = c² for some c > 0, then prove

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(x-a)^2 + (y-b)^2 = c^2

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Question of 277

(x-a)^2 + (y-b)^2 = c^2

Gujarat GsebGSEB Higher Secondary Certificate (HSC) Examination 2025Subjective· 4mImportance★★★★★est

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You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

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Q.If $(x-a)^2 + (y-b)^2 = c^2$ for some $c > 0$ , then prove that $\dfrac{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}}{\dfrac{d^2y}{dx^2}}$ is independent of $a$ and $b$ .

Q.If $(x-a)^2 + (y-b)^2 = c^2$ for some $c > 0$ , then prove that $\dfrac{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}}{\dfrac{d^2y}{dx^2}}$ is independent of $a$ and $b$ .

Q.If (x−a)2+(y−b)2=c2(x-a)^2 + (y-b)^2 = c^2(x−a)2+(y−b)2=c2 for some c>0c > 0c>0, then prove that [1+(dydx)2]3/2d2ydx2\dfrac{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}}{\dfrac{d^2y}{dx^2}}dx2d2y​[1+(dxdy​)2]3/2​ is independent of aaa and bbb.

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Q.If (x−a)2+(y−b)2=c2(x-a)^2 + (y-b)^2 = c^2(x−a)2+(y−b)2=c2 for some c>0c > 0c>0, then prove that [1+(dydx)2]3/2d2ydx2\dfrac{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}}{\dfrac{d^2y}{dx^2}}dx2d2y​[1+(dxdy​)2]3/2​ is independent of aaa and bbb.

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Q.If $(x-a)^2 + (y-b)^2 = c^2$ for some $c > 0$ , then prove that $\dfrac{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}}{\dfrac{d^2y}{dx^2}}$ is independent of $a$ and $b$ .

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Q.If $(x-a)^2 + (y-b)^2 = c^2$ for some $c > 0$ , then prove that $\dfrac{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}}{\dfrac{d^2y}{dx^2}}$ is independent of $a$ and $b$ .

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