JEE Challenger
More from Conic Sections
MathEasyAdv 2007 • Paper 2

Axis of Symmetry of a Parabola

This question contains Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements.

STATEMENT-1 :\textbf{STATEMENT-1 :} The curve y=−x22+x+1y = \frac{-x^2}{2} + x + 1 is symmetric with respect to the line x=1x = 1.

because\textbf{because}

STATEMENT-2 :\textbf{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

Correct
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

Topics & Concepts

Conic SectionsParabolaAxis of SymmetryQuadratic Curves

Step-by-Step Solution

  1. Complete the square for the curve equation:
    y=−12(x2−2x)+1=−12[(x−1)2−1]+1=−12(x−1)2+32y = -\frac{1}{2}(x^2 - 2x) + 1 = -\frac{1}{2}[(x - 1)^2 - 1] + 1 = -\frac{1}{2}(x - 1)^2 + \frac{3}{2}
    This is the standard equation of a vertical downward parabola (x−1)2=−2(y−32)(x - 1)^2 = -2\left(y - \frac{3}{2}\right) with vertex at (1,32)\left(1, \frac{3}{2}\right) and axis of symmetry x=1x = 1.

  2. A parabola is geometrically symmetric with respect to its axis. Since the axis of this parabola is the line x=1x = 1, the curve is symmetric about x=1x = 1. Both statements are true, and Statement-2 is the correct explanation of Statement-1.

Correct Answer: A

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