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Consider two regions R(1):"points P ar...

Consider two regions
`R_(1):"points P are nearer to (1,0) than to "x=-1.`
`R_(2): "Points P are nearer to (0,0) than to (8,0)"` Find the area of the region common to `R_(1) and R_(2).`

Text Solution

Verified by Experts

The correct Answer is:
`(64)/(3)` sq. units
`R_(1) :` points P(x,y) are nearer to (1,0) than to x=-1
`therefore" "sqrt((x-1)^(2)+y^(2))lt|x+1|`
`rArr" "y^(2)lt4x`
`rArr" point P lie inside parabola "y^(2)=4x`
`R_(2) :` Points P(x,y) are nearer to (0,0) than to (8,0)
`therefore" "|x|lt|x-8|`
`rArr" "x^(2)ltx^(2)-16x+64`
`rArr" "xlt4`
`rArr" "`point P lie to the left side of line x=4
The area of common region of `R_(1) and R_(2)` is area bounded by x=4 and `y^(2)=4x.`

`therefore" Required area "=2overset(4)underset(0)int(4-(y^(2))/(4))dy` (Integrating along x-axis)
`=2[4y-(y^(3))/(12)]_(0)^(4)`
`=2[16-(64)/(12)]`
`(64)/(3)` sq. units
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