# Area bounded by the locus

In xy plane a point P moves in such a way that $\angle APB = 30^\circ$ , where points $A$ and $B$ are $(0,0) (10,0)$ respectively , then area (A) bounded by the locus is :

(a) less than$200 \pi$ sq.units

(b) greater than $100\pi$ sq. units

(c) greater than $25\pi$ sq. units

(d) less than $100\pi$ sq. units

My Approach : I assumed P point as $(x,y)$ and using the condition of $\angle APB = 30^\circ$ and using the slope formula of straight lines I found the locus of P which is quadratic in $x$ and $y$ . And the area of triangle $PAB$ came out as $10y/2$ but I am not able to conclude . I also tried to use geometry as $\angle APB = 30^\circ$ and the locus came out to be circle by which (d) option came but i am not sure about (c) part . please help

• As you noted, the locus is a circle. Since $AB$ is a chord of the circle subtended by a $30^\circ$ angle you can work out the radius. Once you have the radius, you can find the area. – Michael Biro Nov 15 '16 at 17:52
• Radius came out to be 10 , by which (d) option is confirmed , please tell how to check (c) option. – saladi Nov 15 '16 at 17:54
• $100 \pi > 25 \pi$? – Michael Biro Nov 15 '16 at 17:55
• What a terribly flawed set of choices. No matter what the area is, either 2 or 3 of the answers are correct. – Mark Fischler Nov 15 '16 at 18:00