The differential equation of the family of circles with fixed radius r and with the center on the y-axis is:

y^{2} (1 + y_{1}^{2}) = r^{2} y_{1}^{2}

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y^{2} = r^{2} y_{1} + y_{1}^{2}

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x^{2} (1 + y_{1}^{2}) = r^{2} y_{1}^{2}

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x^{2} = r^{2} y_{1} + y_{1}^{2}

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Solution

The correct option is C $x^{2} (1 + y_{1}^{2}) = r^{2} y_{1}^{2}$
$\Rightarrow x^{2} + (y - k)^{2} = r^{2}$
$\Rightarrow 2 x + 2 (y - k) y_{1} = 0$
$\Rightarrow (y - k) = \frac{- x}{y_{1}}$
$\Rightarrow x^{2} + \frac{x^{2}}{y_{1}^{2}} = r^{2}$
$\Rightarrow x^{2} (1 + y_{1}^{2}) = r^{2} y_{1}^{2}$

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The differential equation of the family of circles with fixed radius r and with the center on the y-axis is:

The correct option is C x2(1+y21)=r2y21⇒x2+(y−k)2=r2⇒2x+2(y−k)y1=0⇒(y−k)=−xy1⇒x2+x2y21=r2⇒x2(1+y21)=r2y21

The correct option is C $x^{2} (1 + y_{1}^{2}) = r^{2} y_{1}^{2}$
$\Rightarrow x^{2} + (y - k)^{2} = r^{2}$
$\Rightarrow 2 x + 2 (y - k) y_{1} = 0$
$\Rightarrow (y - k) = \frac{- x}{y_{1}}$
$\Rightarrow x^{2} + \frac{x^{2}}{y_{1}^{2}} = r^{2}$
$\Rightarrow x^{2} (1 + y_{1}^{2}) = r^{2} y_{1}^{2}$