Question

11.
The point which divides the line segment joining the points A (0, 5) and
B (5, 0) internally in the ratio 2:3 is
12.
The pair of lines represented by the equations 2x+y+3 = 0 and 4x+ky+6=
0 will be parallel if value of k is​

Answer 1

Question 1: The point divides the line segment joining the points A(0,5) and B(5,0) internally in the ratio 2:3. Find the co-ordinates of the point.

Solution:

Given points,

A(0,5)

x₁ → 0

y₁ → 5

B(5,0)

x₂ → 5

y₂ → 0

Given that the line is divided by the point P(x,y) in the ratio 2:3, i.e,

m₁ : m₂ = 2 : 3

Using the Section formula, we get,

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

$\Longrightarrow \sf P(x,y) = \Huge \text{(} \dfrac{2(5) + 3(0)}{2 + 3} , \dfrac{2(0) + 3(5)}{2 + 3} \Huge \text{)}$

$\Longrightarrow \sf P(x,y) = \Huge \text{(} \dfrac{10}{5} , \dfrac{15}{5} \Huge \text{)}$

$\Longrightarrow \sf P(x,y) = (2,3)$

∴ The point which divides the line segemt joining the points A(0,5) and B(5,0) internally in the ratio 2:3 has the co-ordinates P(2,3)

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Question 2: The pair of lines represented by the equations 2x + y + 3 = 0 and 4x + ky + 6 = 0 are parallel. Then the value of 'k' is?

Given equations;

2x + y + 3 = 0

4x + ky + 6 = 0

If two lines are parallel,

$\Longrightarrow \ \sf \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

$\Longrightarrow \ \sf \dfrac{2}{4} = \dfrac{1}{k} = \dfrac{3}{6}$

Equating 1/k = 2/4 we get,

$\Longrightarrow \ \sf \dfrac{2}{4} = \dfrac{1}{k}$

$\Longrightarrow \ \sf 2k = 4$

$\Longrightarrow \ \sf k = \dfrac{4}{2}$

$\Longrightarrow \ \sf k = 2$

∴ k = 2

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