Find the equations of two straight lines drawn through the point $(0,a)$ on which the perpendiculars drawn from the point $(2a,2a)$ are each of length $a$.

My Attempt: The equation of line passing through $(0,a)$ is $y=mx+a$ where $m$ is the slope. How do I get the value of $m$?

  • $\begingroup$ The distance between each line you want and point $(2a,2a)$ is $a$. Do you see that? $\endgroup$ – toliveira Jan 7 '18 at 16:47

Well, recall the length of a perpendicular from point $(x_{1},y_{1})$ to the line $ax+by+c=0$ is equal to the distance between them and is thus given by $$\dfrac{|ax_{1}+by_{1}+c|}{\sqrt{a^2+b^2}}$$

So the length of a perpendicular from point $(2a,2a)$ on the line $mx-y+a=0$ would be given by $$\dfrac{|2ma-a|}{\sqrt{m^2+1}}=a$$ So squaring and dividing by $a^2$ we have $$\dfrac{4m^2-4m+1}{m^2+1}=1 \implies m=0 \text{ or } m=\dfrac{4}{3}$$


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