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202212201209 Solution to 2001-CE-AMATH-I-10

Two lines L_1:x+y-5=0 and L_2:2x-3y=0 intersect at a point A. Find the equations of the two lines passing through A whose distances from the origin are equal to 2.

Roughwork.

Solving for (a,b) point of intersection:

\begin{cases} a+b-5 = 0 \\ 2a-3b =0 \\ \end{cases}

we have it A(3,2).

Let the equations of the two lines be

\begin{cases} l_1: & a_1x+b_1y+c_1 = 0 \\ l_2: & a_2x+b_2y+c_2 = 0 \\ \end{cases}

Despite the formula

\displaystyle{\textrm{distance}(ax+by+c=0,(x_0,y_0))=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}}

Wikipedia on Distance from a point to a line

let’s rely on first principles. So draw a picture.

And the rest is left an exercise for the reader.