A **bullet nose curve** is a quartic curve that gets its name because each half of the curve tapers to a pointed shape, just like the tip of a bullet.

The equation for a bullet nose curve, a special case of the Lamé curve, is

x

^{2}y

^{2}– a

^{2}y

^{2}+ b

^{2}x

^{2}= 0.

or, equivalently:

a

^{2}y

^{2}– b

^{2}x

^{2}= x

^{2}y

^{2}or

y = ± bx / √(a

^{2}– x

^{2}).

As a Cartesian parametrization: x = a cos t; y = b cot t.

The curve has three inflection points, where the curve changes from concave up to concave down (or vice versa). The curve has one singularity (called a node), two vertical asymptotes, two axes of symmetry, and three double points.

## Equation for a Tangent Line of a Bullet Nose Curve: Example

**Example question:** Find an equation for the tangent line at (1, 1) for the bullet nose curve y = |x| / √(2 – x^{2}) [1].

**Solution: **

Step 1: **Find the equation for the line.** As x = 1 is positive, |x| becomes x. So the equation is:

y = x / √(2 – x^{2})

Step 2: **Find the derivative of the equation from Step 1.** I used Symbolab’s derivative calculator [2] to get:

Step 3: **Insert the x-value (from the question) into the equation for the first derivative (from Step 2).** At (1, 1), the slope of the tangent line is f′(1) =

= **2**.

Step 4: **Insert your answer from Step 3 into the linear equation y = mx + b:**

y = 2x + b.

Step 5: **Figure out the “b” in the linear equation.** This is where the line crosses the vertical axis. We know that the point (1, 1) lies on the tangent line, so we can substitute that in to find “b”:

- 1 = 2(1) + b.
- 1 = 2 + b
- b = -1

The tangent line crosses the y-axis at y = -1:

This means that the equation is** y = 2x -1.**

## References

Graphs made with Desmos.com.

[1] Frith, R. University of Alaska at Anchorage.Assignment Previewer.

[2] Symbolab solver: First Derivative.