Answer to Question #154114 in Trigonometry for Anne

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.
• When the derivative is considered to estimate the turning points,

$\frac{dy}{dx}=\frac{x}{\sqrt{16+x^2}}=0\\ x=0$

• This means that the curve has a turning point at x=0 simultaneously y = 4 at that point.
• As x reaches infinity y reaches infinity concluding that this curve is has a minimum at (0,4) expands towards y's positive direction.
• Therefore, a tangent or a normal could be drawn at (0,4) not at the origin.
• And as it is easy to figure out, the tangent drawn at (0,4) has a gradient of 0 (according to the derivative at x=0) hence the tangent is parallel to x axis. Since y=4 at x=0, line parallel to x axis at this point is y=4
• Again any normal to the x axis is parallel to the y axis hence at that point, the normal is x=0 turning out to be again the y axis.

• Tangent: y=4
• Normal: x=0/y axis