Construct conic loci and convert coordinate forms

Describe the allowed parametric path

Read the parameter interval together with the Cartesian equation for each line, ellipse, parabola or hyperbola

A locus is a set of points satisfying a condition

A line can be defined by a linear equation whose x and y coefficients are not both zero, or as the perpendicular bisector of two distinct points. A circle is the set at a fixed positive distance from a centre. A parameter or polar relation must specify its allowed values; the equation alone may describe a larger set

A direction vector gives a parametric line

For x=1+2t,y=−2+t,eliminate t to obtain x−2y−5=0. With t∈R this is the whole line. With−1≤t≤2 it is only the closed segment from(−1,−3) to(5,0); with t≥0 it is a ray. Vertical lines work by taking the x direction component0

Parametric segment

\(\displaystyle x=1+2t,\quad y=-2+t,\quad -1\le t\le2\)

For the closed segment, give the endpoint x-coordinate at t=2

Circle and ellipse parametrisations retain an interval

For x=h+a cos t,y=k+b sin t with a,b>0,use cos²t+sin²t=1 to obtain(x−h)²/a²+(y−k)²/b²=1. Equal a,b give a circle. The interval0≤t<2π traces the entire curve once;0≤t≤π traces only its upper half. Coordinate bounds describe a box, not the locus by themselves

Which part is traced by x=1+4cos t,y=−2+3sin t,0≤t≤π?

Give the maximum y-coordinate on that arc

Eliminating a parameter can lose the traced range

For x=1+2t,y=−1+t²,−1≤t≤2,the Cartesian equation is(x−1)²=4(y+1). The actual arc has−1≤x≤5 and−1≤y≤3. Its endpoints are(−1,0),(5,3),but its lowest point(1,−1) occurs at t=0, not at either parameter endpoint

Give the minimum y-coordinate of this restricted arc

Secant and tangent can trace both hyperbola branches

The parametrisation x=1+3sec t,y=−2+4tan t gives(x−1)²/9−(y+2)²/16=1. Use0≤t<2π withπ/2 and3π/2 excluded. Positive secant gives the right branch and negative secant gives the left. At t=π/3, the point is (7,−2+4√3)

Give the x-coordinate at t=π/3 in this parametrisation

Connect path geometry to calculus

A Cartesian equation may describe more points than one restricted parameter interval. A spatial curve additionally needs all three coordinate functions