The points P (0, 2) and Q (3, 7) lie on the line l₁, as shown in Figure 2. The line l₁ is perpendicular to l₂, passes through Q and crosses the x-axis at the point R, as shown in Figure 2. Find a) an equation for l₂, giving your answer in the form ax + by + c = 0, where a, b and c are integers, b) the exact coordinates of R, c) the exact area of the quadrilateral ORQP, where O is the origin.

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The points P (0, 2) and Q (3, 7) lie on the line l₁, as shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 11 - 2016 - Paper 1

Question 11

The points P (0, 2) and Q (3, 7) lie on the line l₁, as shown in Figure 2. The line l₁ is perpendicular to l₂, passes through Q and crosses the x-axis at the point ... show full transcript

Worked Solution & Example Answer:The points P (0, 2) and Q (3, 7) lie on the line l₁, as shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 11 - 2016 - Paper 1

Step 1

a) an equation for l₂, giving your answer in the form ax + by + c = 0, where a, b and c are integers

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Answer

To find the equation of the line l₂, we first determine its slope. Since line l₁ passes through points P (0, 2) and Q (3, 7), we calculate its slope (m₁):

$m₁ = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 2}{3 - 0} = \frac{5}{3}$

The slope of line l₂ (m₂) is the negative reciprocal of the slope of l₁ because the two lines are perpendicular:

$m₂ = -\frac{1}{m₁} = -\frac{3}{5}$

Using point slope form with point Q (3, 7):

$y - 7 = -\frac{3}{5}(x - 3)$

Simplifying this,

$y - 7 = -\frac{3}{5}x + \frac{9}{5}$

$y = -\frac{3}{5}x + \frac{44}{5}$

Now, converting to the standard form:

$3x + 5y - 44 = 0$

Thus, the equation of l₂ is:

$3x + 5y - 44 = 0$ .

Step 2

b) the exact coordinates of R

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Answer

To find the x-intercept (point R) of the line l₂, we set y = 0 in the equation of l₂:

$3x + 5(0) - 44 = 0$

Solving for x:

$3x - 44 = 0 \implies x = \frac{44}{3}$

Therefore, the exact coordinates of R are:

$R \left( \frac{44}{3}, 0 \right)$ .

Step 3

c) the exact area of the quadrilateral ORQP, where O is the origin

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Answer

The vertices of quadrilateral ORQP are O(0, 0), R(\frac{44}{3}, 0), Q(3, 7), and P(0, 2).

Using the shoelace formula for the area:

$Area = \frac{1}{2} | x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) |$

Substituting the coordinates:

$Area = \frac{1}{2} | 0 \cdot 0 + \frac{44}{3} \cdot 7 + 3 \cdot 2 + 0 \cdot 0 - (0 \cdot \frac{44}{3} + 0 \cdot 3 + 2 \cdot 0 + 7 \cdot 0) |$

Calculating this:

$Area = \frac{1}{2} | 0 + \frac{308}{3} + 6 + 0 |$

$= \frac{1}{2} \left( \frac{308}{3} + \frac{18}{3} \right) = \frac{1}{2} \left( \frac{326}{3} \right) = \frac{163}{3}$

Thus, the exact area of quadrilateral ORQP is:

$\frac{163}{3}$ .

The points P (0, 2) and Q (3, 7) lie on the line l₁, as shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 11 - 2016 - Paper 1

Worked Solution & Example Answer:The points P (0, 2) and Q (3, 7) lie on the line l₁, as shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 11 - 2016 - Paper 1

a) an equation for l₂, giving your answer in the form ax + by + c = 0, where a, b and c are integers

b) the exact coordinates of R

c) the exact area of the quadrilateral ORQP, where O is the origin

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