The line l_1 passes through the point (9, -4) and has gradient \frac{1}{3} - Edexcel - A-Level Maths Pure - Question 9 - 2005 - Paper 1

To find the intersection point P of the lines l_1 and l_2, we first determine the equation of l_2 using the point-slope form where it passes through the origin (0,0) and has a gradient of -2:

$y = -2x$

Now we set the equations of l_1 and l_2 equal to find the intersection:

$-2x = \frac{1}{3}x - 7$

Multiplying through by 3 to eliminate the fraction gives:

$-6x = x - 21$

Rearranging this leads to:

To calculate the area of \Delta OCP, we can use the area formula:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

The base OC can be determined from its coordinates O(0,0) and C(0,-7), which gives:

( OC = 7 ).

The height from P(3, -6) to the y-axis (x=0) is the x-coordinate of P, which is 3. Hence,

$\text{Area} = \frac{1}{2} \times OC \times height = \frac{1}{2} \times 7 \times 3 = \frac{21}{2}.$

Thus, the exact area of \Delta OCP is ( \frac{21}{2} ).