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The radius of a circle, having minimum a...

The radius of a circle, having minimum area, which touches the curve `y=4−x^2` and the lines, `y=|x|` is: (a) `4(sqrt2+1)` (b) `2(sqrt2+1)` (c) `2(sqrt2-1)` (d) `4(sqrt2-1)`

Text Solution

Verified by Experts

We can create a rough diagram for the curve `y = 4-x^2` and line `y = |x|`.
Please refer to video for the figure.
From the figure, we can see that x-coordinate of the circle will be `0`.
`:.` Curve `y= 4-x^2` will touch the `y`-axis at `y = 4`.
Let radius of the circle is `r`. Then, center of the circle will be `(0,4-r)`
Also, the distace from the center of the circle to the line `y-x = 0` is `r`.
`:. (4-r-0)/(sqrt(1^2+(-1)^2)) = r`
`=>4-r = sqrt2r`
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