Question

Give the equations of two lines passing through (2, 14). How many more such lines are there, and why?

Answer 1

Hint: Assume that the line has a slope m and write the equation of line using point slope form. Consider all possible values of m to find the number of lines passing through the point (2, 14).

Complete step-by-step answer:
We have to write the equation of two lines passing through the point (2, 14) and count all the lines that can pass through that point.

Let’s assume that the equation of line passing through the point (2, 14) has slope m.

We know that the equation of line passing through the point \[\left( {{x}_{1}},{{y}_{1}} \right)\] and having slope m is \[y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)\].

Substituting \[{{x}_{1}}=2,{{y}_{1}}=14\] in the above equation, we have \[y-14=m\left( x-2 \right)\].

Rearranging the above equation, we have \[mx-y-2m+14=0\].

The general equation of line passing through (2, 14) is \[mx-y-2m+14=0\].

We will now find two lines passing through the point.

Substituting \[m=1\] in the above equation, we have \[x-y-2+14=0\Rightarrow x-y+12=0\].

Substituting \[m=2\] in the above equation, we have \[2x-y-4+14=0\Rightarrow 2x-y+10=0\].

Thus, the two equations of lines passing through the given point is \[x-y+12=0\] and \[2x-y+10=0\].

We will now count all possible lines passing through this point.

We know that the general equation of line passing through the given point is \[mx-y-2m+14=0\].

We can substitute all real values of m to find the lines passing through the point. Each real value of m will give a unique equation of line.

As the size of the set of real numbers is infinite, the number of lines that can pass through the given point is infinite as well.

Hence, an infinite number of lines can pass through the point (2, 14).

Note: To write two equations of lines passing through the given point, we can take any real value of m and substitute it in the general equation of line passing through that point. It’s not necessary to take the above two values of m.