Geometry

2D Coordinate Geometry

2D Coordinate Geometry: Level 5 Challenges

         

There exists 22 points on Cartesian coordinate. Point BB has coordinates (0,1)(0,1). Point AA is on the origin (0,0)(0,0).

Point AA and BB always has a constant distance of 11 from each other.

Point BB moves with a horizontal velocity (parallel to the x-axis) of v.v.

Point AA moves with a horizontal velocity of 2v.2v.

Point AA's vertical velocity (parallel to the y-axis) is 00 while Point BB is allowed to move vertically in order to keep the constant distance of 11.

All this movement is happening in the first quadrant.

Let PP be the area made by the figure defined by: the xx axis, the yy axis, and the path traveled by point BB up to where it meets the xx axis. Find 1000P.\left\lfloor 1000P \right\rfloor .


After you solve this, you might want to try a continuation of this problem.

Try my Other Problems

In the diagram above, ABCD and PQRS are both rectangles. Points P, Q, R, and S lie on segments AB\overline{AB}, BC\overline{BC}, CD\overline{CD}, and DA\overline{DA}, respectively, and BQ<QC\overline{BQ} < \overline{QC}.

If AB=36AB=36 and BC=50BC=50, then the maximum possible value of BQBQ can be written in the form aba-\sqrt{b}, where a,bNa,b \in \mathbb{N}. What is a+ba+b?

Given line x+2y=3x+2y=3 intersects circle x2+y2+x6y+a=0x^2+y^2+x-6y+a=0 at PP and QQ. If OO is the origin and OPOQOP\perp OQ, what is the value of aa?

Let two Rods ABAB of length 1010 and CDCD of length 2020 are sliding on smooth standard co-ordinate axis such that their ends are always con-cyclic. If the locus of centre of that circle which pass through all four points A,B,C,DA,B,C,D is expressed as:

(axby)2+(bxay)2=c2 (ax-by)^2 + (bx-ay)^2 = c^2

For positive integers a,b,ca,b,c. What is the minimum value of a+b+ca+b+c

Details and Assumptions

  • Diagram not up to scale
Inspired from Kushal Patankar's Problem

Two lines pass through the point (6,7)(-6, 7) and each is a distance of 2 from the origin at their closest to the origin. What is the sum of the slopes of these lines?

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