Question

Consider the following equations.

x = 1 − t^2, y = t − 5, −2 ≤ t ≤ 2

Eliminate the parameter to find a Cartesian equation of the curve.

(blank) for for −7 ≤ y ≤ −3

Answer 1

To find the Cartesian equation of the curve given by the parametric equations x = 1 - t^2 and y = t - 5, solve for t in terms of y and substitute back to get x = -y^2 - 10y - 24.

Step-by-step explanation:

The student's problem involves parametric equations that define a curve where x and y are both expressed as functions of a parameter t. This question requires finding a Cartesian equation for the curve by eliminating the parameter.

To eliminate the parameter t, we can first express t in terms of y from the second equation: y = t - 5, which rearranges to t = y + 5. Substituting this into the first equation x = 1 - t^2, we get:

x = 1 - (y + 5)^2

Now we have a quadratic equation in terms of y:

x(t) = 1 - (y + 5)^2 = 1 - (y^2 + 10y + 25)

x(t) = -y^2 - 10y - 24

Therefore, the Cartesian equation is x = -y^2 - 10y - 24 suitable for the given range for y, which is -7 ≤ y ≤ -3.

Answer 2

Step-by-step explanation:

`y=t-5\\t=y+5`

Taking the value of t and puting this value into the first equation, we got

`x=1-(y+5)^(2)`