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Geometry

# 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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