If the area of the circle $4{ x }^{ 2 }+4{ y }^{ 2 }+8x-16y+\lambda =0$ is $9\pi$ sq. units, then the value of $\lambda$ is

Dividing the circle's equation by 4 gives x squared plus y squared plus 2x minus 4y plus lambda over 4 equals zero. The radius formula for a circle is the square root of g squared plus f squared minus c, so here the radius is the square root of 1 squared plus 2 squared minus lambda over 4. Because the area is 9 pi, the radius is 3, which makes the squared radius 9. Setting the expression equal to 9 gives 1 plus 4 minus lambda over 4 equals 9, so lambda equals -16.

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