A circle touches the y-axis at the point (0,4 | vedclass

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(C) The equation of a circle touching the $y$-axis at $(0,4)$ is $(x-0)^2 + (y-4)^2 + \lambda x = 0$.Since it passes through $(2,0)$,we substitute $x=

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$4x + 3y - 8 = 0$

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(C) The equation of a circle touching the $y$-axis at $(0,4)$ is $(x-0)^2 + (y-4)^2 + \lambda x = 0$.Since it passes through $(2,0)$,we substitute $x=2$ and $y=0$:$(2-0)^2 + (0-4)^2 + \lambda(2) = 0$ $\Rightarrow 4 + 16 + 2\lambda = 0$ $\Rightarrow 2\lambda = -20$ $\Rightarrow \lambda = -10$.Thus,the circle equation is $x^2 + (y-4)^2 - 10x = 0$,which simplifies to $x^2 + y^2 - 10x - 8y + 16 = 0$.The center is $(5,4)$ and the radius $r = \sqrt{5^2 + 4^2 - 16} = \sqrt{25 + 16 - 16} = 5$.$A$ line $ax + by + c = 0$ is a tangent if the perpendicular distance from the center $(5,4)$ to the line equals the radius $5$.For $A: |3(5) - 4(4) - 24| / \sqrt{3^2 + (-4)^2} = |15 - 16 - 24| / 5 = |-25| / 5 = 5$ (Tangent).For $B: |3(5) + 4(4) - 6| / \sqrt{3^2 + 4^2} = |15 + 16 - 6| / 5 = |25| / 5 = 5$ (Tangent).For $C: |4(5) + 3(4) - 8| / \sqrt{4^2 + 3^2} = |20 + 12 - 8| / 5 = |24| / 5 = 4.8 
eq 5$ (Not a tangent).For $D: |4(5) - 3(4) + 17| / \sqrt{4^2 + (-3)^2} = |20 - 12 + 17| / 5 = |25| / 5 = 5$ (Tangent).

$A$ circle touches the $y$-axis at the point $(0,4)$ and passes through the point $(2,0)$. Which of the following lines is not a tangent to this circle?

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