Geometry

# Locus of Points

Given $$A=(5,0)$$ and $$B=(0,4)$$, what is the equation of the locus of points $$P$$ such that $\lvert \overline{PA} \rvert^2- \lvert \overline{PB} \rvert^2=9?$

Given $$A=(9,-1)$$ and $$B=(-9,5)$$, what is the equation of the locus of points $$P$$ such that $\lvert \overline{AP} \rvert: \lvert \overline{BP} \rvert=1:2 ,$ where $$\lvert \overline{AP} \rvert$$ denotes the length of $$\overline{AP}?$$

Let $$A=(6,-5)$$ and $$B=(-6,1)$$ be two vertices of triangle $$ABC.$$ What is the locus of vertex $$C$$ such that the centroid of $$\triangle ABC$$ moves along the line $$2x+5y=1?$$

Let $$A=(-6,0)$$, $$B=(3,-3)$$ and $$C=(5,3)$$. What is the equation of the locus of points $$P$$ such that $\lvert \overline{AP} \rvert^2+\lvert \overline{BP} \rvert^2=2 \lvert \overline{CP} \rvert^2?$

If $$A=(-3,0)$$ and $$B=(12,0),$$ what is the equation for the locus of points $$P$$ such that $$\lvert \overline{PA} \rvert =\lvert \overline{PB} \rvert,$$ where $$\lvert \overline{PA} \rvert$$ denotes the length of $$\overline{PA}?$$

×