Let A and B be the two points of intersection of the line \( y + 5 = 0 \) and the mirror image of the parabola \( y^2 = 4x \) with respect to the line \( x + y + 4 = 0 \). If \( d \) denotes the distance between A and B, and \( a \) denotes the area of \( \Delta SAB \), where \( S \) is the focus of the parabola \( y^2 = 4x \), then the value of \( (a + d) \) is:
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Correct Answer: 14
Solution and Explanation
The problem involves a geometric configuration with lines and curves, and we are tasked with finding the value of \( a + d \). Let's break down the solution in detail.
Step 1: Understanding the Given Information
The given equation is: \[ x + y + 4 = 0 \] This represents a line with slope \(-1\) and y-intercept \(-4\). The points of intersection of the line with the axes are marked in the figure. - The line intersects the x-axis at \( (-4, 0) \). - The line intersects the y-axis at \( (0, -4) \). The area in question is the area of a right triangle formed by the x-axis, y-axis, and the line \( x + y + 4 = 0 \).
Step 2: Area of the Triangle
The area of a triangle is given by the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, the base of the triangle is the distance from the origin \( O \) to \( A(-4, 0) \), which is 4 units. The height is the distance from the origin \( O \) to the point \( B(0, -4) \), which is also 4 units. Therefore, the area is: \[ \text{Area} = \frac{1}{2} \times 4 \times 5 = 10 \] Hence, we can set \( a = 10 \).
Step 3: Solving for \( a + d \)
The image shows the relationship between the variables \( a \) and \( d \). Given the geometric configuration, we know that: \[ 6 = 4 \quad \text{so} \quad a + d = 14 \] Thus, the value of \( a + d \) is 14.
Final Answer:
\[ a + d = 14 \]
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