Question

The ratio in which the join of (2, 4) and (10, 12) is divided by the line x = 7 is ______.

Options

• 3 : 5

• 5 : 3

• 1 : 5

• 3 : 1

Answer 1

The ratio in which the join of (2, 4) and (10, 12) is divided by the line x = 7 is 5 : 3. 

Explanation:

Any point on the line \[x = 7\], can be defined as \[(7, y).\]

Let point \[(7, y)\] divide line joining \[(2, 4)\] and \[(10, 12)\] in ratio \[k : 1.\]

By section-formula,

\[(x, y) = \left(\frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2}\right)\]

Substituting values we get:

⇒ \[(7, y) = \left(\frac{k \times 10 + 1 \times 2}{k + 1}, \frac{k \times 12 + 1 \times 4}{k + 1}\right)\]

⇒ \[(7, y) = \left(\frac{10k + 2}{k + 1}, \frac{12k + 4}{k + 1}\right)\]

⇒ \[7 = \frac{10k + 2}{k + 1}\]

⇒ \[7(k + 1) = 10k + 2\]

⇒ \[7k + 7 = 10k + 2\]

⇒ \[10k - 7k = 7 - 2\]

⇒ \[3k = 5\]

⇒ \[k = \frac{5}{3}\]

Substituting value of \[k\] in \[k : 1\], we get:

⇒ \[\frac{5}{3} : 1\]

⇒ \[5 : 3\]