Question

How many times does the line y = -6 intersect with the circle x + y = 6

Answer 1

The line
`y=-6` never intersects with the circle
`x^(2) +y^(2) =6`

Explanation:

The equation of a circle with center
`(h,k)` and radius of
`r` units is given by:

`(x-h)^(2) +(y-k)^(2) =r^(2)`

So, for the circle:

`x^(2) +y^(2) =6`

`h=0\\k=0\\r=√(6)\approx 2.54`

Which is a circle centered at the origin and radius of 2.45.

The line
`y=-6` is a horizontal line with a constant value of -6. Hence, the line never intersect the circle.

I attached you the graphs.