Question

COORDINATE GEOMETRY
CLASS 10
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•Please answer the question in the picture.
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THANKS​

Answer 1

We've been given a point M(11, y) that lies between the line joining the points P(15, 5) and Q(9, 20).

We are asked to find the value of y and the ratio in which M(11, y) divides the line segment joining the points P(15, 5) and Q(9, 20). The ratio in which the point divides the line can be found out by using the Section formula.

$\sf \Longrightarrow M(11, y) = \Bigg\{\dfrac{m_1x_2 + m_2x_1 }{m_1 + m_2} , \dfrac{m_1y_2 + m_2y_1 }{m_1 + m_2} \Bigg \}$

Where the x-coordinate of M is (m₁x₂ + m₂x₁)/(m₁ + m₂) and the y-coordinate of M is (m₁y₂ + m₂y₁)/(m₁ + m₂)

x₁ = 15

x₂ = 9

y₁ = 5

y₂ = 20

$\sf \Longrightarrow M(11, y) = \Bigg\{\dfrac{m_1(9) + m_2(15)}{m_1 + m_2} , \dfrac{m_1(20) + m_2(5) }{m_1 + m_2} \Bigg \}$

$\sf \Longrightarrow M(11, y) = \Bigg\{\dfrac{9m_1 + 15m_2}{m_1 + m_2} , \dfrac{20m_1 + 5m_2}{m_1 + m_2} \Bigg \}$

We know that the x-coordinate 11 is equal to (9m₁ + 15m₂)/(m₁ + m₂). Therefore:

$\sf \Longrightarrow 11 = \dfrac{9m_1 + 15m_2}{m_1 + m_2}$

$\sf \Longrightarrow 11(m_1 + m_2) = 9m_1 + 15m_2$

$\sf \Longrightarrow 11m_1 + 11m_2 = 9m_1 + 15m_2$

$\sf \Longrightarrow 11m_1 - 9m_1 = 15m_2 - 11m_2$

$\sf \Longrightarrow 2m_1 = 4m_2$

$\sf \Longrightarrow \dfrac{m_1}{m_2} = \dfrac{4}{2}$

$\sf \Longrightarrow \dfrac{m_1}{m_2} = \dfrac{2}{1}$

$\sf \Longrightarrow m_1:m_2 = 2:1$

∴ The point M(11, y) divides the line PQ in the ratio 2 : 1.

We know that the y-coordinate is equal to (m₁y₂ + m₂y₁)/(m₁ + m₂).

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

We know that m₁ = 2, m₂ = 1, y₂ = 20 and y₁ = 5.

$\sf \Longrightarrow y = \dfrac{(2)(20) + (1)(5)}{2 + 1}$

$\sf \Longrightarrow y = \dfrac{40 + 5}{3}$

$\sf \Longrightarrow y = \dfrac{45}{3}$

$\sf \Longrightarrow y = 15$

∴ The y-coordinate is 15.

Answer 2

We've been given a point M(11, y) that lies between the line joining the points P(15, 5) and Q(9, 20).

We are asked to find the value of y and the ratio in which M(11, y) divides the line segment joining the points P(15, 5) and Q(9, 20). The ratio in which the point divides the line can be found out by using the Section formula.

$\sf \Longrightarrow M(11, y) = \Bigg\{\dfrac{m_1x_2 + m_2x_1 }{m_1 + m_2} , \dfrac{m_1y_2 + m_2y_1 }{m_1 + m_2} \Bigg \}$

Where the x-coordinate of M is (m₁x₂ + m₂x₁)/(m₁ + m₂) and the y-coordinate of M is (m₁y₂ + m₂y₁)/(m₁ + m₂)

x₁ = 15

x₂ = 9

y₁ = 5

y₂ = 20

$\sf \Longrightarrow M(11, y) = \Bigg\{\dfrac{m_1(9) + m_2(15)}{m_1 + m_2} , \dfrac{m_1(20) + m_2(5) }{m_1 + m_2} \Bigg \}$

$\sf \Longrightarrow M(11, y) = \Bigg\{\dfrac{9m_1 + 15m_2}{m_1 + m_2} , \dfrac{20m_1 + 5m_2}{m_1 + m_2} \Bigg \}$

We know that the x-coordinate 11 is equal to (9m₁ + 15m₂)/(m₁ + m₂). Therefore:

$\sf \Longrightarrow 11 = \dfrac{9m_1 + 15m_2}{m_1 + m_2}$

$\sf \Longrightarrow 11(m_1 + m_2) = 9m_1 + 15m_2$

$\sf \Longrightarrow 11m_1 + 11m_2 = 9m_1 + 15m_2$

$\sf \Longrightarrow 11m_1 - 9m_1 = 15m_2 - 11m_2$

$\sf \Longrightarrow 2m_1 = 4m_2$

$\sf \Longrightarrow \dfrac{m_1}{m_2} = \dfrac{4}{2}$

$\sf \Longrightarrow \dfrac{m_1}{m_2} = \dfrac{2}{1}$

$\sf \Longrightarrow m_1:m_2 = 2:1$

∴ The point M(11, y) divides the line PQ in the ratio 2 : 1.

We know that the y-coordinate is equal to (m₁y₂ + m₂y₁)/(m₁ + m₂).

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

We know that m₁ = 2, m₂ = 1, y₂ = 20 and y₁ = 5.

$\sf \Longrightarrow y = \dfrac{(2)(20) + (1)(5)}{2 + 1}$

$\sf \Longrightarrow y = \dfrac{40 + 5}{3}$

$\sf \Longrightarrow y = \dfrac{45}{3}$

$\sf \Longrightarrow y = 15$

∴ The y-coordinate is 15.