Parametric Equations (Calculus Question)

Question

A curve C is given by parametric equations x = sect and y= tant where 0 ≤ t < pi/2.

a.) Convert the parametric equations to an algebraic equation in rectangular form an state the range of vales from x and y.

b.) Sketch the graph of C points and label all relevant points.

Answer 1

X = Sec(t) and Y = Tan(t) where 0 ≤ t < pi/2

Now if we use Trigonometric identity 1 + (Tan(t))^2 = (Sec(t))^2 then we get:

1 + Y^2= X^2 => X^2 - Y^2 = 1 which is a Hyperbola whose center is (0,0); a= 1 and b = 1

This hyperbola is symmetric across both x-axis and y-axis and it opens up across both (+)ve x-axis

and (-)ve x-axis.

I have no scope to sketch this graph ; Any one can sketch this graph using a TI 84 graphing calculator.

Before graphing, break the hyperbola into 2 parts y = SQRT(X^2 - 1) and y = - SQRT(X^2 - 1)

Answer 2

I cannot sketch the graph in this environment but part a is as follows:

x = sec(t) y = tan(t) we know that tan²(t) + 1 = sec²(t)

x² = sec²(t) y² = tan²(t) then y² + 1 = x²

after rearranging, x² - y² = 1

x varies from +-1 to +- infinity

y varies from 0 to +- infinity

The graph is a hyperbola that opens up in the +x & -x directions with a & b = 1. It is symmetric about both the x & y axis.