Question

AB is a line segment with A = (2, 4) and B = (6, 12). Point P lies on the line segment AB so that P = (3, x), then the ratio AP : PB is ______.

Options

• 3 : 2

• 2 : 3

• 3 : 1

• 1 : 3

Answer 1

AB is a line segment with A = (2, 4) and B = (6, 12). Point P lies on the line segment AB so that P = (3, x), then the ratio AP : PB is 1 : 3.

Explanation:

Let ratio in which \[P\] divides \[AB\] be \[k : 1.\]

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:

⇒ \[(3, x) = \left(\frac{k \times 6 + 1 \times 2}{k + 1}, \frac{k \times 12 + 1 \times 4}{k + 1}\right)\]

⇒ \[(3, x) = \left(\frac{6k + 2}{k + 1}, \frac{12k + 4}{k + 1}\right)\]

⇒ \[\frac{6k + 2}{k + 1} = 3\]

⇒ \[6k + 2 = 3(k + 1)\]

⇒ \[6k + 2 = 3k + 3\]

⇒ \[6k - 3k = 3 - 2\]

⇒ \[3k = 1\]

⇒ \[k = \frac{1}{3}\]

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

⇒ \[\frac{1}{3} : 1\]

⇒ 1 : 3