Line of a proper pencil through a point

Text

Consider the proper pencil of lines $y=mx+3$, with centre $C(0,3)$. Find the line of the pencil passing through the point $P(2,7)$.

Solution

Impose passage through $P(2,7)$ by substituting its coordinates into the pencil equation: $7=m\cdot 2+3\;\Rightarrow\; 2m=4\;\Rightarrow\; m=2.$ The required line is $y=2x+3$. $\boxed{\,m=2,\quad y=2x+3\,}$