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2D Coordinate Geometry

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

Distance between point and line

         

If point \((p,0)\) is equidistant from the two lines \[9x-y+1=0, x-9y-5=0\] and \(p>0\), what is \(p?\)

Consider point \(A=(a,2)\) with \(a > 0\) such that over all points \(P\) on the line \[\frac{x}{4}+\frac{y}{3}=1,\] the minimum value of \(\overline{AP}\) is \(2.\) What is the value of \(3a?\)

Given a point \(P\) on the positive \(x\)-axis such that the distance between \(P\) and the line \(4x+3y+6=0\) is \(10,\) what are the coordinates of \(P ?\)

If real numbers \(\ a\ \) and \(\ b\ \) satisfy \( a^2 + b^2 = 49 \), what is the distance between the lines \[ ax + by = 1\;\;\; \textrm{and} \;\;\; ax + by = 225 ?\]

If point \((p,0)\) is equidistant from the two lines \[12x-y+1=0, x-12y-4=0\] and \(p>0\), what is \(p?\)

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