# Uniqueness of a line

• Jan 23rd 2008, 12:15 AM
sfitz
Uniqueness of a line
The question is:

(a,b) and (c,d) are two distinct points in R^2. Prove there exists a unique line passing through them.

I proved that a line exists:
y=[(d-b)/(c-a)]x + [b-(d-b)a/(c-a)]

But how do I prove that it is unique? I know the technique, you say that there are two and prove that they are the same, but how do I go about it in this problem? Thank you for your help!
• Jan 23rd 2008, 01:34 AM
mr fantastic
Quote:

Originally Posted by sfitz
The question is:

(a,b) and (c,d) are two distinct points in R^2. Prove there exists a unique line passing through them.

I proved that a line exists:
y=[(d-b)/(c-a)]x + [b-(d-b)a/(c-a)]

But how do I prove that it is unique? I know the technique, you say that there are two and prove that they are the same, but how do I go about it in this problem? Thank you for your help!

Assume that there are at least 2 lines:

\$\displaystyle y = m_1 x + c_1\$ and \$\displaystyle y = m_2 x + c_2\$.

Sub the given points into each:

\$\displaystyle y = m_1 x + c_1\$:

\$\displaystyle b = a m_1 + c_1\$ .... (1)
\$\displaystyle d = c m_1 + c_1\$ .... (2)

Solve simultaneously for \$\displaystyle m_1\$ and \$\displaystyle c_1\$.

\$\displaystyle y = m_2 x + c_2\$:

\$\displaystyle b = a m_2 + c_2\$ .... (3)
\$\displaystyle d = c m_2 + c_2\$ .... (4)

Solve simultaneously for \$\displaystyle m_2\$ and \$\displaystyle c_2\$.

Therefore establish that \$\displaystyle m_1 = m_2\$ and \$\displaystyle c_1 = c_2\$.
• Jan 28th 2008, 07:52 PM
ThePerfectHacker
Quote:

Originally Posted by mr fantastic
Assume that there are at least 2 lines:

\$\displaystyle y = m_1 x + c_1\$ and \$\displaystyle y = m_2 x + c_2\$.
.

This does not prove it. What happens if the lines are vertical? Then we cannot write y=mx+b. Instead we should use ax+by = c. This handles all possible lines.