STATEMENT-1 : The curve y = -x²/2 + x + 1 is symmetric with respect to the line x = 1. because STATE

Q: Wait, can't I just use the formula $x = -b/2a$ to find the axis of symmetry? Why go through the trouble of completing the square?

You absolutely can! Using $x = -b/2a$ is a fantastic shortcut for competitive exams. However, completing the square is a fundamental skill that reveals the vertex form of the parabola, which gives you both the axis and the vertex coordinates $(h, k)$ simultaneously. Think of it as understanding the 'DNA' of the curve rather than just memorizing a formula.

Q: How do I know if the axis of symmetry is $x=h$ or $y=k$?

Great question! Look at the squared term. If $x$ is squared (like $(x-h)^2$), the parabola opens vertically, so the axis is the vertical line $x=h$. If $y$ is squared (like $(y-k)^2$), it opens horizontally, and the axis is the horizontal line $y=k$. Always look for the variable that is not squared to determine the direction of opening!

Q: Is Statement-2 really the reason for Statement-1? It feels like a general fact, not a specific explanation.

This is the classic Assertion-Reasoning trap! Statement-2 provides the geometric principle (the 'why') that allows us to conclude Statement-1. Without the knowledge that parabolas are symmetric about their axis, we wouldn't know that finding the axis is sufficient to prove symmetry. Thus, it is the logical foundation.

Question

STATEMENT-1 : The curve $y = \frac{-x^2}{2} + x + 1$ is symmetric with respect to the line $x = 1$. because STATEMENT-2 : A parabola is symmetric about its axis.

Options:

(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True.

Sigma Percentile · Accepted Answer

### Analyzing the Geometry of Symmetry

Welcome, future engineers! Today, we are going to unravel a beautiful problem that tests not just your algebraic skills, but your geometric intuition. We are looking at a quadratic curve and asking: is it symmetric about a specific line? And more importantly, why?

### Phase 1: The Universal Truth

Let's start with Statement-2: "A parabola is symmetric about its axis." This is a fundamental geometric truth.

Imagine a parabola as a perfectly balanced U-shaped curve. If you were to place a mirror along its central axis, the reflection would perfectly overlap with the other side.

Because this is a universal property of all parabolas, we can confidently declare that **Statement-2 is True**.

### Phase 2: The Algebraic Transformation

Now, let's turn our attention to the curve given in Statement-1: $y = -\frac{x^2}{2} + x + 1$. To understand its symmetry, we need to see its "true face"—the standard form.

The equation is a quadratic in $x$, which confirms it is indeed a parabola. To find its axis of symmetry, we need to find its vertex $(h, k)$.

Let's start by clearing the fraction. Multiplying the entire equation by $-2$, we get:
$$-2y = x^2 - 2x - 2$$

Now, let's isolate the $x$ terms. Moving the constant $-2$ to the left, we have:
$$x^2 - 2x = -2y + 2$$

This is where the magic happens. We want to complete the square for $x^2 - 2x$. The coefficient of $x$ is $-2$; half of that is $-1$, and squaring it gives us $1$.

Adding $1$ to both sides:
$$x^2 - 2x + 1 = -2y + 2 + 1$$

This simplifies beautifully to:
$$(x - 1)^2 = -2y + 3$$

Factoring out the $-2$ on the right side, we arrive at the standard form:
$$(x - 1)^2 = -2\left(y - \frac{3}{2}\right)$$

### Phase 3: The Vertex and the Axis

By comparing this to the standard form $(x - h)^2 = 4a(y - k)$, we can instantly see that the vertex $(h, k)$ is $(1, \frac{3}{2})$.

The axis of symmetry for a parabola of this form is the vertical line passing through the vertex, which is $x = h$. Substituting our value of $h$, we get $x = 1$.

This confirms that the curve is indeed symmetric about the line $x = 1$. Thus, **Statement-1 is True**.

### Phase 4: The Logical Synthesis

We have established that Statement-1 is true and Statement-2 is true. Is Statement-2 the correct explanation for Statement-1?

Absolutely. We proved Statement-1 by finding the axis of symmetry, and we were only able to do that because we knew the geometric property that a parabola is symmetric about its axis.

Statement-2 provided the necessary logical framework to validate the symmetry of the curve in Statement-1. Therefore, **Statement-2 is the correct explanation**.

Animated Solution for Mathematics - Conic Sections: STATEMENT-1 : The curve $y = \frac{- x ^{2}}{2} + x + 1$ is symmetric with respect to the line $x = 1$ . because STATEMENT-2 : A parabola is symmetric about its axis.

Select Answer:

Visualized Solution

Analyzing the Statements

Geometry of a Parabola

The Given Curve

Clearing the Fraction

Isolating $x$ Terms

Perfect Square Setup

Standard Form of Parabola

Vertex of the Parabola

Axis of Symmetry

Conclusion for Statement-1

Connecting the Statements

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Geometry of Symmetry

Phase 1

Phase 2

Phase 3

Phase 4

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