Answer to Question #153493 in Calculus for Pia

Question

Find the Tangent and Normal line to the given curve. y=√(16+x^2) at the origin

Answer 1

Solution

• At x=0 y=4: means that the curve does not pass through the origin.
• The curve is a parabola having a minimum at (0,4) towards infinite y.
• Therefore, a tangent & a normal could be drawn at (0,4) not at the origin (0,0).
• Considering the derivative of the curve,

$\qquad\qquad\,\,\,\,\,\,\frac{dy}{dx}=\frac{1}{2\sqrt{16+x^2}}(2x)=\frac{x}{\sqrt{16+x^2}}\\ \qquad\qquad\frac{dy}{dx}_{(0,4)}=0$

• Therefore, tangent is a line having zero gradient $\to$ parallel to the x axis$\to$ y= 4
• Then the normal line is parallel to the y axis$\to$ x=0$\to$ the y axis.

• Tangent: y=4
• Normal: x=0