Write as a Vector Equality x^2+y^2=25 , (x^2)/(5^2)-(y^2)/(5^2)=1

Problem

x2+y2=25,(x2)/5−(y2)/5=1

Solution

1. Identify the first equation as a circle with radius r=5

2. Parametrize the circle using the trigonometric identities x=r*cos(t) and y=r*sin(t)

3. Write the vector equation for the circle as (r_1)*(t)=<5*cos(t),5*sin(t)>

4. Identify the second equation as a hyperbola with semi-axes a=5 and b=5

5. Parametrize the hyperbola using the hyperbolic identities x=a*cosh(t) and y=b*sinh(t)

6. Write the vector equation for the hyperbola as (r_2)*(t)=<5*cosh(t),5*sinh(t)>

Final Answer

(r_1)*(t)=<5*cos(t),5*sin(t)>,(r_2)*(t)=<5*cosh(t),5*sinh(t)>

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