Question

For the curve $2x^2+y^2-4x+2y=1,$ Which of the following(s) is/are true ?

• A. Tangent at point $(1,1)$ is parallel to $x-$axis
• B. Tangent at point $(1,1)$ is parallel to $y-$axis
• C. Normal at point $(1,1)$ is parallel to $x-$axis
• D. Normal at point $(1,1)$ is parallel to $y-$axis

Answer 1

Answer: (A), (D)

Normal at point $(1,1)$ is parallel to $y-$axis

Given curve : $2x^2+y^2-4x+2y=1$
Differentiating both sides w.r.t. $x$
$\Rightarrow 4x+2y{y}^{\prime }-4+2y^{\prime }=0$
$\Rightarrow y^{\prime }=\frac{2(1-x)}{1+y}=\frac{dy}{dx}$
$\Rightarrow$ Slope of tangent at $(1,1)$
$\frac{dy}{dx}{|}_{(1,1)}=\frac{2(1-1)}{1+1}=0$
$\therefore \frac{dy}{dx}{|}_{(1,1)}=0$ (i.e. parallel to $x-$axis)
so, normal will be parallel to $y-$axis.