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2D Coordinate Geometry

In the 1600s, René Descartes married algebra and geometry to create the Cartesian plane.

Level 5

         

There exists \(2\) points on Cartesian coordinate. Point \(A\) has coordinates \((0,1)\). Point \(B\) is on the origin \((0,0)\).

Point \(A\) and \(B\) always has a constant distance of \(1\) from each other.

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

Point \(B\) moves with a horizontal velocity of \(2v\)

Point \(B\)'s vertical velocity (parallel to the y-axis) is \(0\) while Point \(A\) is allowed to move vertically in order to keep the constant distance of \(1\).

All this movement is happening in the first quadrant.

Given that the area made with the \(x\) and \(y\) axis and the path traveled by Point \(A\) is \(P\), find \[\left\lfloor 1000P \right\rfloor \]


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 \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), and \(\overline{DA}\), respectively, and \(\overline{BQ} < \overline{QC}\).

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

Given line \(x+2y=3\) intersects circle \[x^2+y^2+x-6y+a=0\] at \(P\) and \(Q\). If \(O\) is the origin and \(OP\perp OQ\), what is the value of \(a\)?

Let two Rods \(AB\) of length \(10\) and \(CD\) of length \(20\) 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 \(A,B,C,D\) is expressed as:

\[ (ax-by)^2 + (bx-ay)^2 = c^2 \]

For positive integers \(a,b,c\). What is the minimum value of \(a+b+c\)

Details and Assumptions

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

Two lines pass through the point \((–6, 7)\) 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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