Problem 70 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(P\) is a point which moves in the \(x y\) plane such that the point \(P\) is nearer to the centre of a square than any of the sides. The four vertices of the square are \((\pm a, \pm a)\). The region in which \(P\) will move is bounded by parts of parabola of which one has the equation (A) \(y^{2}=a^{2}-2 a x\) (B) \(y^{2}=a^{2}+2 a x\) (C) \(x^{2}=a^{2}-2 a y\) (D) \(x^{2}=a^{2}+2 a y\)

Step-by-Step Solution

Answer

The parabola has the equation \(y^{2}=a^{2}-2ax\) (Option A).

Step 1: Understanding the Problem

The point \(P\) must remain closer to the center \((0, 0)\) of a square rather than the sides of the square. The square has vertices \((\pm a, \pm a)\). The task is to determine the equation of one boundary of the region in which point \(P\) can move.

Step 2: Distance from the Center to Sides of the Square

The center of the square is at \((0, 0)\), and the sides of the square are situated at \(x = \pm a\) and \(y = \pm a\). The perpendicular distance from the center to any side is \(a\). Point \(P\) must be closer to \((0, 0)\) than these horizontal or vertical distances \(a\).

Step 3: Establishing the Distance Condition

The condition to be nearer to the center than the sides implies that the point \((x, y)\) is within a boundary defined by a parabola equidistant from \((0, 0)\) and \(x = a\). For the left side boundary for example, the boundary is closer to the center.

Step 4: Deriving the Equation of the Parabola

Using the focus \((0, 0)\) and directrix \(x = a\), the equation of the parabola is obtained using the formula \((x - a)^2 = 4p(x + p)\), where \(p\) is the perpendicular distance from the focus to the directrix. Solving for \(P(x, y)\), the correct form is \(y^2 = a^2 - 2ax\).

Step 5: Confirm the Correct Option

From Step 4, we have derived the equation \(y^2 = a^2 - 2ax\). Comparing with the options provided, the correct choice is Option (A).

Key Concepts

Parabola

A parabola is a U-shaped curve that you often encounter in mathematics. It is a special set of points, each of which is at an equal distance from a fixed point known as the "focus," and a fixed line called the "directrix." Understanding the structure of a parabola is crucial in solving problems related to coordinate geometry, especially when it involves distances and boundaries.

In our exercise, the parabola is one where its focus is at the origin \(0,0\) because this is our center of the square, and its directrix is a vertical line. This particular setup means the parabola's equation is derived from reflecting over a vertical axis.

To recognize the parabola’s equation, remember these features:

• The canonical form is \(y^2 = 4px\), where \(p\) is the distance from the focus to the directrix.
• Our derived equation \(y^2 = a^2 - 2ax\) showcases how the parabola encompasses regions where the point P remains closer to the center.

Considering the square's layout, the parabola serves to depict the area closer to the center than any side.

Distance Formula

The distance formula is a magical tool in coordinate geometry. It helps calculate the exact length between two points in the plane. This formula is essential when you need to determine the shorter or longer path between geometrical figures.

The standard distance formula is expressed as:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Where \(x_1,y_1\) and \(x_2,y_2\) are coordinates of the two related points.

In the context of the square problem, applying the distance formula aids us in comparing distances between the point \((x, y)\) on the boundary of the region and our center \((0, 0)\), as opposed to any side at \(x = a\) or \(y = a\). This comparison guarantees that point \(P\) remains within the correct region since its distance to \(x = a\) describes part of the boundary condition.

Equidistant Condition

The equidistant condition is a fundamental element when working within coordinate geometry to determine geometric shapes or restrict movement. This condition ensures that a point in a plane maintains an equal distance between a focus and a line or another point.

For our problem, the equidistant condition is pivotal. It sets up the requirement that the point \((x,y)\) stays nearer to the center of the square than its boundaries. This constraint leads us to describe part of the permissible area as a parabola:

• In simpler words, point \((x, y)\) adheres to the rule: It is closer to the origin than to the line \(x = a\).

This rule logically translates into a parabolic form, thereby establishing the exact boundary of the region in which point \(P\) may lie. Through the distance formula and the properties of the parabola, we derive and confirm our equation \(y^2 = a^2 - 2ax\) for one of these parabolic boundaries.