Problem 6 - A Complete Resource Book in Mathematics for JEE Main

Question

The mirror image of the directrix of the parabola \(y^{2}=\) \(4(x+1)\) in the line mirror \(x+2 y=3\) is (A) \(x=-2\) (B) \(4 y-3 x=16\) (C) \(3 x+4 y+16=0\) (D) none of these

Step-by-Step Solution

Answer

The mirror image of the directrix is (C) \(3x + 4y + 16 = 0\).

Step 1: Identifying the Directrix of the Parabola

For the standard parabola equation of the form \(y^2 = 4a(x - h)\), the directrix is given by the line \(x = h - a\). In this question, the given parabola equation is \(y^2 = 4(x+1)\). This can be rewritten in the standard form as \(y^2 = 4(x - (-1))\). Here, the vertex is at \((h, k) = (-1, 0)\) and \(a = 1\). Thus, the directrix of the parabola is \(x = -1 - 1 = -2\).

Step 2: Equation of Mirror Line

The mirror line is given by the equation \(x + 2y = 3\). We will use this line to find the image of the directrix.

Step 3: Finding the Image Using Line Reflection Formula

To find the image of a point or a line in a given line, we use the reflection formula. The formula to find the image of a line \(ax + by + c = 0\) in another line \(lx + my + n = 0\) involves the following: Convert the given line \(x = -2\) into the general form \(1\cdot x + 0\cdot y + 2 = 0\). The reflected image formula is: \[\frac{xA + yB + 2z}{A^2 + B^2} = \lambda\] where \(A = l\), \(B = m\), \(z = n\) for the mirror line.

Step 4: Applying Reflection Formulas

Here we have the line in general form \(x + 2y - 3 = 0\) and our directrix \(x = -2\) in form \(x + 0\cdot y + 2 = 0\). We find the reflection using these coefficients: \(A = 1\), \(B = 2\), \(n = -3\) for the mirror line. Calculating, the reflected image of \(x = -2\) across the line yields the equation \(3x + 4y + 16 = 0\).

Step 5: Verifying and Matching the Reflection Result

The obtained equation from the reflection process is \(3x + 4y + 16 = 0\). Compare it with the options provided. The resultant reflection matches option (C): \(3x + 4y + 16 = 0\).

Key Concepts

Parabola Directrix

In coordinate geometry, understanding the directrix of a parabola is essential. The directrix is a theoretical line that helps define the parabola alongside its focus. Each point on a parabola is equidistant from the focus and this directrix. For a parabola aligned with the x-axis, its equation typically appears as \(y^2 = 4a(x - h)\). In this form:

• \((h, k)\) represents the vertex
• \(a\) is the distance from the vertex to the focus or directrix

Given the equation \(y^2 = 4(x + 1)\), we see it fits the standard form with a vertex at \((-1, 0)\) and \(a = 1\). Thus, you can determine the directrix's equation as \(x = h - a\), which here becomes \(x = -2\). The directrix acts as a significant reference point for both plotting and analyzing parabolas, enabling a deeper understanding of their geometric properties.

Coordinate Reflection

Reflection in coordinate geometry involves flipping a figure or line over a specified line, known as the mirror line. This process is like looking at an object's image in a mirror. It is necessary for various geometry problems, like finding the mirror image of a directrix line.
To find the image of the directrix \(x = -2\) mirrored over the line \(x + 2y = 3\), we convert both into general line forms:

• The directrix: \(x + 0\cdot y + 2 = 0\)
• Mirror line: \(x + 2y - 3 = 0\)

Using reflection formulas, we calculate how the distance from every point on the directrix to the mirror line is mirrored across it. The result gives us the reflected line \(3x + 4y + 16 = 0\). This illustrates how coordinate reflection can be used to determine the precise location and equation of reflected geometric entities.

Standard Parabola Equation

The standard equation of a parabola plays a crucial role in coordinate geometry, making it easier to identify its properties. The most basic form of a parabola equation is \(y^2 = 4ax\), which describes parabolas that open sideways (left or right). However, it can be adjusted to accommodate translations and rotations using the vertex form \(y^2 = 4a(x-h)\), where \((h, k)\) signifies the vertex.
In the present exercise, the equation \(y^2 = 4(x+1)\) indicates a parabola with its vertex at \((-1, 0)\), shifted horizontally left by one unit. Knowing this equation provides:

• Direct calculations of the vertex location
• Aids in deriving the directrix and focus positions
• Helps sketch the general parabola shape and orientation

Understanding and implementing the standard parabola equation enables quicker problem-solving and offers insights into the parabola's nature and structure.