Question

The point on the curve$x^2+y^2=a^2,y\ge 0$ at which the tangent is parallel to the $x$-axis is

• A. $(a,0)$
• B. $(-a,0)$
• C. $\left(\frac{a}{2},\sqrt{\frac{3}{2}}a\right)$
• D. $(0,a)$
• E. $(0,a^2)$

Answer 1

The correct option is D

$(0,a)$

Explanation for the correct answer:

Slope of tangent at any point :

Given, $x^2+y^2=a^2$

The first order derivative of the equation of a curve at a point, gives the slope of the tangent to the curve at the point

Differentiating the equation of the curve with respect to $x$ we get

$2x+2y\left(\frac{dy}{dx}\right)=0$

$\Rightarrow$$\frac{dy}{dx}=\frac{-x}{y}$

It is given that the tangent is parallel to the $x$ axis

The $x$ axis has slope$=0$ and parallel lines have equal slope

Hence, the slope of the tangent is $0$

Now $\frac{dy}{dx}=0$

$\Rightarrow$ $x=0$

Substituting the value of $x=0$ in the equation of the curve we get,

$\Rightarrow$ $y^2=a^2$

$\Rightarrow$ $y=a$

Hence the point on the given curve at which the tangent is parallel to the $x$ axis is $(0,a).$

Hence, option D is the correct answer.