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2D Coordinate Geometry

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

Level 4

         

What is the area enclosed by \((y-x)^{2}=4\) and \((y+x)^{2}=4?\)

The points \((3, 7), (6, 2),\) and \((2, k)\) are the vertices of a triangle. For how many real values of \(k\) is the triangle a right triangle?

Tangents are drawn from \(P (6,8)\) to the circle \(x^2+y^2=r^2\). Find the radius of the circle such that the area of the triangle formed by tangents and chord of contact is maximum.

Triangle \( ABC\) has coodinates \(A= (-4, 0)\), \(B= (4 , 0)\), and \(C= (0 , 3)\).

Let \(P\) be the point in the first quadrant such that \(\triangle ABP\) has half the area of \(\triangle ABC\) but both triangles have the same perimeter.

What is the length of \(CP\)? If your solution is in a form of \(\sqrt{d}\), submit \(d\) as the answer.

\(ABC\) is an equilateral triangle such that vertices \(B,C\) lie on two parallel lines at a distance of 6 units. If \(A\) lies between the parallel lines at a distance 4 units from one of them, then the length of the side of the triangle is of the form

\(\large{A\sqrt{\frac{B}{C}}}\), where \(A,B,C\) are co prime natural numbers.

Find \(A+B+C\).

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