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 3 - 2016 - Paper 1

Question 3

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 3 - 2016 - Paper 1

Step 1

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 line l2, we first calculate the slope of line l1 using points P and Q.

The coordinates for P are (0, 2) and Q are (3, 7). The slope (m) of line l1 is given by:

$m_{l1} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 2}{3 - 0} = \frac{5}{3}$

Since l2 is perpendicular to l1, the slope of l2 (m_{l2}) is the negative reciprocal of m_{l1}:

$m_{l2} = -\frac{3}{5}$

Using point Q (3, 7) and the slope, we can apply the point-slope form of the line equation:

$y - y_1 = m_{l2}(x - x_1)$

Substituting in the values:

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

Expanding this:

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

To eliminate the fraction, we can multiply through by 5:

$5y - 35 = -3x + 9$

Rearranging gives:

$3x + 5y - 44 = 0$

Thus, the equation for l2 in the required form is:

$3x + 5y - 44 = 0$

Step 2

the exact coordinates of R

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Answer

To find the coordinates of R, we set y = 0 in the equation of line l2:

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

This simplifies to:

$3x - 44 = 0$

Solving for x:

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

Thus, the coordinates of R are:

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

Step 3

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

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Answer

To find the area of quadrilateral ORQP, we can divide it into two triangles: ORQ and OPQ.

For triangle ORQ, the vertices are O(0, 0), R(\frac{44}{3}, 0), and Q(3, 7). The area (A) can be calculated using the formula:

$A = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) |$

Substituting the coordinates we have:

$A_{ORQ} = \frac{1}{2} | 0(0-7) + \frac{44}{3}(7-0) + 3(0-0) | = \frac{1}{2} | \frac{308}{3} | = \frac{154}{3}$

For triangle OPQ, the vertices are O(0, 0), P(0, 2), and Q(3, 7).

Applying the same area formula:

$A_{OPQ} = \frac{1}{2} | 0(2-7) + 0(7-0) + 3(0-2) | = \frac{1}{2} | 3(-2) | = 3$

Now, the total area of quadrilateral ORQP is:

$A_{ORQP} = A_{ORQ} + A_{OPQ} = \frac{154}{3} + 3 = \frac{154}{3} + \frac{9}{3} = \frac{163}{3}$

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 3 - 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 3 - 2016 - Paper 1

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

the exact coordinates of R

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

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