Question

The ratio in which the point p(3/4, 5/12) divides the segmet joining the points A (1/2, 3/4) and B ( 2,-5) is:​

Answer 1

Correction: The ratio in which the point P(3/4, 5/12) divides the segment joining the points A(1/2, 3/2) and B (2, -5) is:​

We've been given points A(1/2, 3/2) and B(2, -5), which when joined, are divided by the point P(2, -5).

We have to find the ratio in which the P(2, -5) divides AB. We can find the ratio using the Section formula.

$\rm \Longrightarrow P(x, y) = \Bigg\{\dfrac{m_1x_2 + m_2x_1}{m_1 + m_2} , \dfrac{m_1y_2 + m_2y_1}{m_1 + m_2} \Bigg\}$

Also:

$\rm \Longrightarrow x = \dfrac{m_1x_2 + m_2x_1}{m_1 + m_2}$

$\rm \Longrightarrow y = \dfrac{m_1y_2 + m_2y_1}{m_1 + m_2}$

Where m₁ : m₂ is the ratio in which the line is divided.

Here:

x₁ = 1/2

x₂ = 2

y₁ = 3/4

y₂ = -5

$\rm \Longrightarrow P\Bigg\{\dfrac{3}{4} , \dfrac{5}{12}\Bigg\} = \Bigg\{\dfrac{m_1(2) + m_2\big(\frac{1}{2}\big)}{m_1 + m_2} , \dfrac{m_1(-5) + m_2\big(\frac{3}{2}\big)}{m_1 + m_2} \Bigg\}$

$\rm \Longrightarrow P\Bigg\{\dfrac{3}{4} , \dfrac{5}{12}\Bigg\} = \Bigg\{\dfrac{2m_1 + \big(\frac{m_2}{2}\big)}{m_1 + m_2} , \dfrac{-5m_1 + \big(\frac{3m_2}{2}\big)}{m_1 + m_2} \Bigg\}$

We know that the x-coordinate of P(3/4, 5/12) which is 3/4, is equal to (2m₁ + (m₂/2))/(m₁ + m₂).

Similarly the y-coordinate, which is 5/12, is equal to (-5m₁ + (3m₂/2))/(m₁ + m₂).

$\rm \Longrightarrow \dfrac{3}{4} = \dfrac{2m_1 + \big(\frac{m_2}{2}\big)}{m_1 + m_2}$

$\rm \Longrightarrow \dfrac{3}{4} = \dfrac{\bigg(\dfrac{4m_1 + m_2}{2}\bigg)}{m_1 + m_2}$

$\rm \Longrightarrow \dfrac{3}{4} = \dfrac{4m_1 + m_2}{2(m_1 + m_2)}$

$\rm \Longrightarrow \dfrac{3}{4} = \dfrac{4m_1 + m_2}{2m_1 + 2m_2}$

$\rm \Longrightarrow 3(2m_1 + 2m_2) = 4(4m_1 + m_2)$

$\rm \Longrightarrow 6m_1 + 6m_2 = 16m_1 + 4m_2$

$\rm \Longrightarrow 6m_2 - 4m_2= 16m_1 - 6m_1$

$\rm \Longrightarrow 2m_2= 10m_1$

$\rm \Longrightarrow \dfrac{2}{10} = \dfrac{m_1}{m_2}$

$\rm \Longrightarrow \dfrac{1}{5} = \dfrac{m_1}{m_2}$

$\rm \Longrightarrow \bold{\dfrac{m_1}{m_2} = \dfrac{1}{5}}$

Now, Let us equate 5/12, with (-5m₁ + (3m₂/2))/(m₁ + m₂).

$\rm \Longrightarrow \dfrac{5}{12} = \dfrac{-5m_1 + \big(\frac{3m_2}{2}\big)}{m_1 + m_2}$

$\rm \Longrightarrow \dfrac{5}{12} = \dfrac{\bigg(\dfrac{-10m_1 + 3m_2}{2}\bigg)}{m_1 + m_2}$

$\rm \Longrightarrow \dfrac{5}{12} = \dfrac{-10m_1 + 3m_2}{2(m_1 + m_2)}$

Transposing 2 to the other side we get:

$\rm \Longrightarrow \dfrac{5 \times 2}{12} = \dfrac{-10m_1 + 3m_2}{m_1 + m_2}$

$\rm \Longrightarrow \dfrac{5}{6} = \dfrac{-10m_1 + 3m_2}{m_1 + m_2}$

$\rm \Longrightarrow 5(m_1 + m_2) = 6(-10m_1 + 3m_2)$

$\rm \Longrightarrow 5m_1 + 5m_2 = -60m_1 + 18m_2$

$\rm \Longrightarrow 5m_1 + 60m_1 = 18m_2 - 5m_2$

$\rm \Longrightarrow 65m_1 = 13m_2$

$\rm \Longrightarrow \dfrac{m_1}{m_2} = \dfrac{13}{65}$

$\rm \Longrightarrow \bold{\dfrac{m_1}{m_2} = \dfrac{1}{5}}$

Therefore, the line AB is divided in the ratio 1:5 by the point P(3/4, 5/12).