Understanding the Concept of Locus

In mathematics, the locus of a point is the set of all points (a curve, surface, or line) whose location satisfies a given condition or set of conditions. When asked to find the locus of a point, our goal is to derive an algebraic equation that describes this path. The general strategy involves:

1. Assigning general coordinates (typically $(x, y)$) to the point whose locus we need to find.
2. Using the given geometric conditions to establish relationships between these coordinates and any other moving points or fixed points.
3. Eliminating any intermediate or auxiliary variables to obtain a single equation solely in terms of $(x, y)$.

Key Formulas and Concepts for this Problem

1. Midpoint Formula: If a point $M(x, y)$ is the midpoint of a line segment connecting two points $P(x_1, y_1)$ and $Q(x_2, y_2)$, its coordinates are given by: $x = \frac{x_1 + x_2}{2}$ $y = \frac{y_1 + y_2}{2}$
2. Equation of a Parabola: A point $(x_0, y_0)$ lies on the parabola ${y^2} = 8x$ if and only if its coordinates satisfy the equation, i.e., ${y_0}^2 = 8x_0$. This equation defines the set of all points that constitute the parabola.

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Step-by-Step Solution

1. Define the Given Points and the Locus Point

Let's systematically label the points involved in the problem:

• We are given a fixed point $P = (1, 0)$. This point does not move.
• Let $Q$ be a moving point on the parabola ${y^2} = 8x$. Since $Q$ moves along the parabola, its coordinates are not fixed. We denote them using general variables, say $(x_Q, y_Q)$.
  • Why this step is important: Clearly defining variables helps avoid confusion. Since $Q$ lies on the parabola, its coordinates must satisfy the parabola's equation. This gives us our first fundamental relationship: ${y_Q}^2 = 8x_Q \quad \text{(Equation 1)}$
• Let $M$ be the midpoint of the line segment $PQ$. We are asked to find the locus of $M$. By convention, we assign its general coordinates as $(x, y)$.
  • Why this step is important: The coordinates $(x, y)$ are what we want to find an equation for. This equation will describe the path traced by $M$ as $Q$ moves along the parabola.

2. Apply the Midpoint Formula to Relate M, P, and Q

Now, we use the midpoint formula to establish a connection between the coordinates of our locus point $M(x, y)$ and the coordinates of the fixed point $P(1, 0)$ and the moving point $Q(x_Q, y_Q)$.

• For the x-coordinate of $M$: $x = \frac{x_Q + x_P}{2} \implies x = \frac{x_Q + 1}{2} \quad \text{(Equation 2)}$

• For the y-coordinate of $M$: $y = \frac{y_Q + y_P}{2} \implies y = \frac{y_Q + 0}{2} \implies y = \frac{y_Q}{2} \quad \text{(Equation 3)}$

• Why this step is important: These two equations are crucial because they establish the direct link between the coordinates of our locus point $(x, y)$ and the coordinates of the moving point $(x_Q, y_Q)$ on the parabola. Our ultimate goal is to eliminate $x_Q$ and $y_Q$.

3. Express Coordinates of Q in terms of M's Coordinates

Our goal is to find an equation involving only $x$ and $y$. To achieve this, we need to eliminate the auxiliary variables $x_Q$ and $y_Q$ from Equation 1 (the parabola's equation). We can do this by rearranging Equations 2 and 3 to express $x_Q$ and $y_Q$ in terms of $x$ and $y$.

• From Equation 2, multiply both sides by 2: $2x = x_Q + 1$ Now, isolate $x_Q$: $x_Q = 2x - 1 \quad \text{(Equation 4)}$

• From Equation 3, multiply both sides by 2: $y_Q = 2y \quad \text{(Equation 5)}$

• Why this step is important: This is a pivotal step in any locus problem. By isolating $x_Q$ and $y_Q$, we are preparing to substitute these expressions into Equation 1. This substitution will eliminate the coordinates of $Q$ and leave us with an equation solely in terms of the coordinates of $M$.

4. Substitute into the Parabola's Equation

Now, we use the fundamental condition that $Q(x_Q, y_Q)$ lies on the parabola ${y^2} = 8x$. We substitute the expressions for $x_Q$ (from Equation 4) and $y_Q$ (from Equation 5) into Equation 1 (${y_Q}^2 = 8x_Q$):

$(2y)^2 = 8(2x - 1)$

• Why this step is important: This substitution is the heart of finding the locus. We are essentially saying: "Since $Q$ must satisfy ${y_Q}^2 = 8x_Q$, and since $x_Q$ and $y_Q$ are directly related to $x$ and $y$ as shown in Equations 4 and 5, then the coordinates $(x, y)$ of $M$ must satisfy this new equation." This effectively translates the condition on $Q$ to a condition on $M$.

5. Simplify to Find the Locus Equation

Finally, we perform the necessary algebraic simplification to obtain the equation relating $x$ and $y$:

$4y^2 = 16x - 8$

To simplify further and put it into a standard form, we can divide the entire equation by 4:

$\frac{4y^2}{4} = \frac{16x}{4} - \frac{8}{4}$ $y^2 = 4x - 2$

Now, rearrange the terms to match the format typically seen in options or for standard forms of conic sections, usually setting one side to zero:

$y^2 - 4x + 2 = 0$

• Why this step is important: This simplified equation is the intrinsic relationship that must be satisfied by the coordinates $(x, y)$ of any point $M$ that is the midpoint of $PQ$. Therefore, this equation represents the locus of $M$.

Comparing this result with the given options, we find that it matches option (A).

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Tips and Common Mistakes to Avoid

• Variable Management: Be meticulous with variable names. Clearly distinguish between fixed points, moving points on a curve, and the point whose locus is being found. A common convention is:

  • Fixed points: Specific coordinates (e.g., $(1,0)$).
  • Moving point on a curve: $(x_Q, y_Q)$ or $(x_1, y_1)$.
  • Locus point: $(x, y)$. This clarity prevents errors during substitution.

• Purpose of Substitution: Always remember that the primary goal in locus problems is to eliminate the auxiliary variables (like $x_Q, y_Q$ here) that describe the intermediate moving point, leaving an equation purely in terms of the locus point's coordinates $(x, y)$. Every algebraic manipulation should lead towards this goal.

• Algebraic Accuracy: Locus problems often involve straightforward but critical algebraic manipulations. Double-check all calculations, especially when squaring terms, distributing multiplication, or simplifying fractions. A small error can lead to an incorrect locus equation and choosing the wrong option.

• Parameterization (Alternative Approach - Highly Recommended for JEE): For curves like parabolas, it is often more efficient and less prone to algebraic errors to parameterize the coordinates of the moving point. For the parabola ${y^2} = 8x$ (which is of the form ${y^2} = 4ax$ with $a=2$), a point $Q$ can be represented as $(at^2, 2at)$, or specifically $(2t^2, 4t)$.

  Let's re-solve using parameterization:

  1. Let the fixed point be $P(1, 0)$.
  2. Let the moving point $Q$ on the parabola ${y^2} = 8x$ be parameterized as $(2t^2, 4t)$. (Here, $a=2$ for ${y^2}=4ax$)
  3. Let $M(x, y)$ be the midpoint of $PQ$. Using the midpoint formula: $x = \frac{x_Q + x_P}{2} = \frac{2t^2 + 1}{2} \quad \text{(Equation A)}$ $y = \frac{y_Q + y_P}{2} = \frac{4t + 0}{2} = 2t \quad \text{(Equation B)}$
  4. Now, the goal is to eliminate the parameter $t$. From Equation B, express $t$ in terms of $y$: $t = \frac{y}{2}$
  5. Substitute this expression for $t$ into Equation A: $x = \frac{2\left(\frac{y}{2}\right)^2 + 1}{2}$ $x = \frac{2\left(\frac{y^2}{4}\right) + 1}{2}$ $x = \frac{\frac{y^2}{2} + 1}{2}$
  6. Simplify the complex fraction by multiplying the numerator and denominator by 2: $x = \frac{y^2 + 2}{4}$
  7. Rearrange to get the locus equation: $4x = y^2 + 2$ $y^2 - 4x + 2 = 0$ This method often simplifies the elimination process, especially when the curve has a standard parametric form, making it a powerful tool for JEE problems.

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Summary and Key Takeaway

The locus of the midpoint of $PQ$ is another parabola described by the equation ${y^2} - 4x + 2 = 0$. This problem perfectly illustrates the standard methodology for finding loci:

1. Identify and Label: Clearly define the fixed point, the moving point on a given curve, and the point whose locus is sought (using $(x,y)$).
2. Formulate Relationships: Use geometric properties (like the midpoint formula) to relate the locus point's coordinates to the moving point's coordinates. Also, use the equation of the given curve to define the moving point's coordinates.
3. Eliminate Auxiliary Variables: This is the most critical step. Express the coordinates of the moving point in terms of the locus point's coordinates, and substitute these into the equation of the given curve. Alternatively, use parametric equations for the moving point and eliminate the parameter.
4. Simplify: Perform algebraic simplifications to obtain the final equation in terms of $(x, y)$, which is the desired locus. Understanding this systematic approach is key to solving a wide variety of locus problems in coordinate geometry.