Question

Statement I: the curve $y=-\frac{x^2}{2}+x+1$ is symmetric with respect to the line $x=1$.

Statement II: A parabola is symmetric about its axis.

• A. Statement I is correct, Statement II is correct; Statement II is a correct explanation for Statement I
• B. Statement I is correct, Statement II is correct; Statement II is not correct explanation for Statement I
• C. Statement I is correct, Statement II is incorrect
• D. Statement I is incorrect, Statement II is correct

Answer 1

The correct option is A

Statement I is correct, Statement II is correct; Statement II is a correct explanation for Statement I

Explanation for the correct option:

Finding the value of $x$:

Given equation is,

$y=-\frac{x^2}{2}+x+1$

and this expression is symmetric with respect to the line $x=1$

We can write the given equation as,

$$\begin{gathered} y=-\frac{x^2}{2}+x+1 \\ \Rightarrow y=\frac{-x^2+2x+2}{2} \\ \Rightarrow 2y=-x^2+2x+2 \\ \Rightarrow x^2-2x-2=-2y \\ \Rightarrow x^2-2x-3+1=-2y\left[\because -3+1=-2\right] \\ \Rightarrow x^2-2x+1=-2y+3 \\ \Rightarrow x^2-x-x+1=-2\left(y-\frac{3}{2}\right) \\ \Rightarrow x\left(x-1\right)-x\left(x-1\right)=-2\left(y-\frac{3}{2}\right) \\ \Rightarrow \left(x-1\right)\left(x-1\right)=-2\left(y-\frac{3}{2}\right) \\ \Rightarrow {\left(x-1\right)}^2=-2\left(y-\frac{3}{2}\right) \end{gathered}$$

Clearly, this is a parabola with the axis

$$\begin{gathered} x-1=0 \\ x=1 \end{gathered}$$

Hence given curve is symmetric about x-axis

Therefore, the correct answer is option (A).