JEE Main & Advanced Mathematics Straight Line Question Bank Self Evaluation Test - Straight Lines & Pair Striaght Lines

The area of the region bounded by the locus of a point P satisfying \[d(P,A)=4\], where A is (1, 2) is

A) 64 sq. unit

B) 54 sq. unit

C) \[\,16\pi \,\,sq.\text{ }unit\]

D) None of these

Correct Answer: A

Solution :

[a] We have, max \[\{\left| x-1 \right|,\left| y-2 \right|\}=4\]

If \[\{\left| x-1 \right|\ge \left| y-2 \right|\},\]

then \[\left| x-1 \right|=4,\]

i.e., if \[(x+y-3)(x-y+1)\ge 0,\]

Then \[x=-3or5,\]

If \[\left| y-2 \right|\ge \left| x-1 \right|,\]

Then \[\left| y-2 \right|=4\]

i.e., \[(x+y-3)(x-y+1)\le 0,\]

Then \[y=-2or6.\]

So, the locus of P bounds a square, the equation of whose sides are \[x=-3,x=5,y=-2,y=6\]

Thus, the area is \[{{(8)}^{2}}=64.\]