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2D Coordinate Geometry

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

Angle Bisector

Which of the following is one of the equations of the bisectors of the angles between \[\begin{align} 12x+3y+11 &= 0 \text{ and} \\ 3x-12y &= 0 ? \end{align} \]

Which of the following lines is a locus of equidistant points from the two lines \[\begin{align} 12x+5y+31 &= 0 \text{ and} \\ -5x+12y+3 &= 0 ? \end{align} \]

For the three points \(P=(-4,0),\) \(Q=(0,0)\) and \(R=(2,2\sqrt{3}),\) what is the equation of the bisector of the angle \(\angle{PQR}?\)

Let \(\displaystyle y=-\frac{1}{\sqrt{5}}x\) be one of the two bisectors of the angles between two lines \[px+qy+r=0, p'x+q'y+r'=0\] which intersect at the point \(P=(-2\sqrt{5},2).\) What is the \(y\)-intercept of the other bisector?

Let \(P=(-5,0)\), \(Q=(0,0)\) and \(R=(5,5\sqrt{3})\) be three points on the \(xy\)-plane. Then what is the equation of the bisector of the angle \(\angle PQR ?\)

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