October 1st, 2012, 02:00 AM  #1 
Member Joined: Oct 2012 Posts: 30 Thanks: 0  find the area of the region
i got trouble of this question, anyone can help? The figure shows a shaded region consisting of all points inside a square (with sides of 2 cm) that are closer to the centre than to the sides of the square. Find the area of the region. You must show all appropriate working. thanks. 
October 1st, 2012, 02:45 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs  Re: find the area of the region
The locus of the set of points equidistant to a point and a line is by definition a parabola. I would use symmetry to find 1/8 of the area as in the diagram: [attachment=0:2lsnpr1j]area001.jpg[/attachment:2lsnpr1j] We have placed the center of the square at the origin. The square in the diagram is the upper right quadrant of the entire square. The parabola with its focus at the origin and its directrix as the line is: This intersects the line in the first quadrant for Hence, the requested area is: 
October 1st, 2012, 04:01 AM  #3 
Member Joined: Oct 2012 Posts: 30 Thanks: 0  Re: find the area of the region
i'm sorry but can i ask how you get the x=sqrt{6}2 thanks. 
October 1st, 2012, 08:19 AM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs  Re: find the area of the region Take the positive root, by the quadratic formula: 
October 1st, 2012, 03:46 PM  #5 
Member Joined: Oct 2012 Posts: 30 Thanks: 0  Re: find the area of the region
thanks so much!!

October 2nd, 2012, 08:56 AM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs  Re: find the area of the region
I made an error in my computation above, specifically in finding the equation of the parabola. It should be: Equating this to , we find the intersection is at: And so the area A in question is: 
October 3rd, 2012, 12:05 AM  #7 
Member Joined: Oct 2012 Posts: 30 Thanks: 0  Re: find the area of the region
so did you divide that by 4 in stand of 8?

October 3rd, 2012, 12:12 AM  #8 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs  Re: find the area of the region
I still divided the region into 8 equal portions, but the denominator for the parabolic curve was 2 instead of 4, so the outside constant is 4 instead of 2.

October 3rd, 2012, 04:32 AM  #9 
Newbie Joined: Oct 2012 Posts: 4 Thanks: 0  Re: find the area of the region
Sorry, but how did you get the equation y = (1x^2)/2? I tried using the distance formula and equated it with y and I got y = (x^2 + 1)/2?

October 3rd, 2012, 04:48 AM  #10 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs  Re: find the area of the region
We want the point(s) (x,y) equidistant from the point (0,0) and the line , hence: 

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