Question

Find the area of the region that comprises all points that satisfy two conditions x${}^2$ + y${}^2$ + 6x + 8y $\le$0 and 4x$\ge$3y

• A.
  
• B.
  
• C.
  
• D.
  

Answer 1

The correct option is C

x${}^2$+y${}^2$+6x+8y<0
x${}^2$ + 6x + 9 - 9 + y${}^2$ + 8y + 16 - 16 < 0
(x+3)${}^2$ + (y+4)${}^2$<25
This region is a circle with centre (-3,-4) with radius 5.
Substituting x=y=0 we get the inequation satisfied. The circle also passes through origin.
The line 4x=3y passes through (-3,-4) which is the diameter of the circle.
Therefore the area required is the area of semicircle
Therefore required area = $\frac{25\pi }{2}$