The points A (8, -4) and B (-1, 3) are the endpoints of the line segment [AB]. Find the coordinates of the point C, which divides [AB] internally in the ratio 4 : 1. The line l has a slope of m and contains the point (q, r), where m, q, r ∈ ℝ are all positive. Find the co-ordinates of the point where l cuts the y-axis, in terms of m, q, and r. The line k has a slope of -2. The line j makes an angle of 30° with k. Find one possible value of the slope of the line j. Give your answer in the form d + e√f, where d, e, f ∈ ℤ.

Leaving Cert Mathematics Co-Ordinate Geometry: The points A (8, -4) and B (-1,

The points A (8, -4) and B (-1, 3) are the endpoints of the line segment [AB] - Leaving Cert Mathematics - Question 2 - 2022

Question 2

The points A (8, -4) and B (-1, 3) are the endpoints of the line segment [AB]. Find the coordinates of the point C, which divides [AB] internally in the ratio 4 : 1... show full transcript

Worked Solution & Example Answer:The points A (8, -4) and B (-1, 3) are the endpoints of the line segment [AB] - Leaving Cert Mathematics - Question 2 - 2022

Step 1

Find the coordinates of the point C, which divides [AB] internally in the ratio 4 : 1.

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Answer

To find the coordinates of point C, we use the section formula:

$C = \left( \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}, \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \right)$

Here, the coordinates of A are (8, -4) and the coordinates of B are (-1, 3). The ratio is 4:1, so:

For x-coordinates:

$C_x = \frac{4 \cdot (-1) + 1 \cdot 8}{4 + 1} = \frac{-4 + 8}{5} = \frac{4}{5}$

For y-coordinates:

$C_y = \frac{4 \cdot 3 + 1 \cdot (-4)}{4 + 1} = \frac{12 - 4}{5} = \frac{8}{5}$

Thus, the coordinates of point C are ( \left( \frac{4}{5}, \frac{8}{5} \right) ).

Step 2

Find the co-ordinates of the point where l cuts the y-axis, in terms of m, q, and r.

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Answer

The equation of the line l in slope-intercept form is:

$y = mx + c$

To find the y-intercept (where the line cuts the y-axis), we set x = 0:

$y = r - mq$

So, the coordinates where the line l cuts the y-axis are (0, r - mq).

Step 3

Find one possible value of the slope of the line j.

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Answer

The tangent of the angle θ between two lines is given by:

$\tan(\theta) = \frac{m_2 - m_1}{1 + m_1 m_2}$

Given:

m₁ = -2 (slope of line k)
θ = 30°

We know that: $\tan(30°) = \frac{1}{\sqrt{3}}$

Let m₂ be the slope of line j. Then,

$\frac{1}{\sqrt{3}} = \frac{m_2 + 2}{1 - 2m_2}$

Cross-multiplying gives:

$1 - 2m_2 = \sqrt{3}(m_2 + 2)$

Solving this equation leads to:

$m_2 = 8 + 5\sqrt{3}$

Thus, one possible value of the slope of line j is ( 8 + 5\sqrt{3} ).

The points A (8, -4) and B (-1, 3) are the endpoints of the line segment [AB] - Leaving Cert Mathematics - Question 2 - 2022

Worked Solution & Example Answer:The points A (8, -4) and B (-1, 3) are the endpoints of the line segment [AB] - Leaving Cert Mathematics - Question 2 - 2022

Find the coordinates of the point C, which divides [AB] internally in the ratio 4 : 1.

Find the co-ordinates of the point where l cuts the y-axis, in terms of m, q, and r.

Find one possible value of the slope of the line j.

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