The line $L_1$ has equation $4y + 3 = 2x$ - Edexcel - A-Level Maths Pure - Question 1 - 2011 - Paper 2

First, we need to find the slope of $L_1$ by rearranging its equation:

[ 4y = 2x - 3 \Rightarrow y = \frac{1}{2}x - \frac{3}{4} ] The slope of $L_1$ is $\frac{1}{2}$. Thus, the slope of $L_2$, being perpendicular, will be the negative reciprocal:

[ m_{L_2} = -2 ]

Using point-slope form, the equation of $L_2$ through the point $(2, 4)$ is:

[ y - y_1 = m(x - x_1) ] [ y - 4 = -2(x - 2) ] [ y - 4 = -2x + 4 ] [ y = -2x + 8 ] Arranging this into the standard form: [ 2x + y - 8 = 0 ] The equation of $L_2$ is $2x + y - 8 = 0$.

To find the intersection point $D$ of $L_1$ and $L_2$, we substitute the equation of $L_2$ into $L_1$:

From $L_1$: [ 4y + 3 = 2x \Rightarrow y = \frac{2x - 3}{4} \] Substituting into the equation of $L_2$: [ 2x + \frac{2x - 3}{4} - 8 = 0 \] Multiplying by 4 to eliminate the fraction: [ 8x + 2x - 3 - 32 = 0 \Rightarrow 10x = 35 \Rightarrow x = 3.5 \] Now substituting $x$ back into one of the original equations to find $y$: [ y = \frac{2(3.5) - 3}{4} = 1.25 \] Thus, the coordinates of the point $D$ are $(3.5, 1.25)$.

To find the length $CD$, we can use the distance formula:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
Substituting the coordinates of $C(2, 4)$ and $D(3.5, 1.25)$:

[ CD = \sqrt{(3.5 - 2)^2 + (1.25 - 4)^2} ]
[ = \sqrt{(1.5)^2 + (-2.75)^2} ]
[ = \sqrt{2.25 + 7.5625} = \sqrt{9.8125} ]
[ = \sqrt{\frac{39}{4}} = \frac{3}{2}\sqrt{5}]

To calculate the area of quadrilateral $ACBE$, we can split it into triangles $ABC$ and $ABE$.

1. Area of Triangle ABC: Using the formula: [ Area = \frac{1}{2} \times base \times height ]
   where base is the length of $AC$ and height is the vertical distance:

   • Length of $AC = 2 - 2 = 0$ (no width here); this will need clarity based on points selected instead.

2. Area of Triangle ABE: Using the same formula, we identify $AB$ and its height from $E$.

This requires calculating the area based on the identified lengths and coordinates:

• Calculate coordinates of point $E$ through ratios based on segment $CD$ for area calculations leading to final area formulation.
• Without specific lengths from segments in context, we would usually utilize:

$(x_1 * y_2 + x_2 * y_3 + ... - [y_1 * x_2 + y_2 * x_3 + ...]) / 2$.