Consider the circles, all of unit radius given by the form (x - h)² + (y - k)² = 1 The radius of curvature at any point (x, y) for these circles may be also expressed as

Correct Answer is : [1+(dydx)2]3=(d2ydx2)2≤ft[ 1 + ≤ft( dy/dx )² ]³ = ≤ft( {d²y}{dx²} )²[1+(dxdy)2]3=(dx2d2y)2

≤ft[ 1 + ≤ft( dy/dx )² ]³ = ≤ft( {d²y}{dx²} )²

≤ft[ 1 + ≤ft( dy/dx )² ]³/2} = ≤ft( {d²y}{dx²} )³/2}

≤ft[ 1 + ≤ft( dy/dx )² ]² = ≤ft( {d²y}{dx²} )³

Correct Answer is : [1+(dydx)2]3=(d2ydx2)2≤ft[ 1 + ≤ft( dy/dx )² ]³ = ≤ft( {d²y}{dx²} )²[1+(dxdy)2]3=(dx2d2y)2

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HCU MCA 2013 Paper

Question : 63 of 75

Marks: +1, -0

^{2}

^{2}

$\left[ 1 + \left( \frac{dy}{dx} \right)^{2} \right]^{3} = \left( \frac{d^{2}y}{dx^{2}} \right)^{2}$
$\left[ 1 + \left( \frac{dy}{dx} \right)^{2} \right]^{3/2} = \left( \frac{d^{2}y}{dx^{2}} \right)^{3/2}$
$\left[ 1 + \left( \frac{dy}{dx} \right)^{2} \right]^{2} = \left( \frac{d^{2}y}{dx^{2}} \right)^{3}$
None of the above

Solution:

(a) Given equation of circle,

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;( x - h )^{2} + ( y - k )^{2} = 1\;\;\;\;\;\;\;\;\;\;\text{...(i)}

On differentiating w.r.t x, we get,

\Rightarrow\;\;\;\;\;\;\;2 ( x - h ) + 2 ( y - k ) \frac{dy}{dx} = 0

\Rightarrow\;\;\;\;\;\;\;\;\;\;( x - h ) + ( y - k ) \frac{dy}{dx} = 0 \;\;\;\;\;\;\;\;\;\;\text{...(ii)}

Again, diffierentiating w.r.t x, we get

\Rightarrow\;1 + ( y - k ) \frac{d^{2}y}{dx^{2}} + \frac{dy}{dx} \cdot \frac{dy}{dx} = 0

\Rightarrow\;\;\;\;\;\;\;\;\;\frac{d^{2}y}{dx^{2}} ( y - k ) + \left( \frac{dy}{dx} \right)^{2} = -1 \;\;\;\;\;\;\;\;\;\;\text{...(iii)}

From Eqs. (ii) and (iii), we obtain

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;( y - K ) = -\frac{1 + \left( \frac{dy}{dx} \right)^{2}}{\frac{d^{2}y}{dx^{2}}}

and

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;( x - h ) = \frac{ \left(1 + \left( \frac{dy}{dx} \right)^{2}\right) \frac{dy}{dx} }{ \frac{d^{2}y}{dx^{2}} }

On substituting these values in the Eq. (i), we get

\;\;\;\;\;\;\;\;\;\;\;\left[ 1 + \left( \frac{dy}{dx} \right)^{2} \right]^{3} = \left( \frac{d^{2}y}{dx^{2}} \right)^{2}

which is the required differential equation.

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