The number of circles touching the line y - x | vedclass

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(D) Let the center of the circle be $(h, k)$ and its radius be $r$. Since the circle touches the $y$-axis,the radius $r = |h|$. Since the circle also

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Infinite

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(D) Let the center of the circle be $(h, k)$ and its radius be $r$. Since the circle touches the $y$-axis,the radius $r = |h|$. Since the circle also touches the line $x - y = 0$,the perpendicular distance from the center $(h, k)$ to the line must be equal to the radius $r$. Using the distance formula,$\frac{|h - k|}{\sqrt{1^2 + (-1)^2}} = |h|$. This simplifies to $|h - k| = |h|\sqrt{2}$. Squaring both sides,$(h - k)^2 = 2h^2$,which gives $h^2 - 2hk + k^2 = 2h^2$,or $h^2 + 2hk - k^2 = 0$. For any given $k 
eq 0$,this quadratic equation in $h$ provides real values for $h$. Since there are infinitely many values for $k$,there are infinitely many such circles.

The number of circles touching the line $y - x = 0$ and the $y$-axis is

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