Geometry

# 2D Coordinate Geometry: Level 4 Challenges

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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