Find a parametrisation of the hyperbola that is obtained by translating the hyperbola with equation
9x^2 - (y^2)/16 =1
by -2 unit(s) right and 4 unit(s) up
x=?
y=?
you know that sec^2(t)-tan^2(t) = 1
so, let
x = sec(t)/3
y = 4tan(t)
oops. That gives the untranslated parameterization. You want to shift things, so
x = sec(t)/3 - 2
y = 4tan(t) + 4
so that
9(x+2)^2 - (y-4)^2/16 = 1
To find the parametrization of the translated hyperbola, we first need to determine the equation of the original hyperbola after translation.
The translation involves shifting the hyperbola 2 units to the right and 4 units up. Therefore, we need to replace x with (x - 2) and y with (y + 4) in the original equation.
Starting with the original equation:
9x^2 - (y^2)/16 = 1
After translation:
9(x - 2)^2 - ((y + 4)^2)/16 = 1
To express the equation in terms of x and y, we can simplify the equation further:
9(x^2 - 4x + 4) - ((y^2 + 8y + 16)/16) = 1
9x^2 - 36x + 36 - (y^2 + 8y + 16)/16 = 1
Multiply both sides by 16 to get rid of the denominator:
144x^2 - 576x + 576 - y^2 - 8y - 16 = 16
Simplifying further:
144x^2 - 576x - y^2 - 8y + 540 = 0
Now, we have obtained the equation of the hyperbola after translation. To find the parametrization, we can parameterize x and y in terms of a parameter t.
Let x = a cos(t) and y = b sin(t), where a and b are positive constants determining the shape and size of the hyperbola.
Substituting these values into the equation, we get:
144(a cos(t))^2 - 576(a cos(t)) - (b sin(t))^2 - 8(b sin(t)) + 540 = 0
Using the trigonometric identity cos^2(t) + sin^2(t) = 1:
144a^2 cos^2(t) - 576a cos(t) - b^2 sin^2(t) - 8b sin(t) + 540 = 0
Rearranging the terms:
(144a^2 cos^2(t)) - (b^2 sin^2(t)) + (-576a cos(t) - 8b sin(t)) + 540 = 0
To simplify the equation further, we can use the trigonometric identity cos^2(t) - sin^2(t) = cos(2t) and sin(2t) = 2sin(t)cos(t):
(144a^2 cos^2(t) - b^2 sin^2(t)) + (-576a cos(t) - 8b sin(t)) + 540 = 0
(144a^2 cos(2t) - b^2 sin(2t)) + (-576a cos(t) - 8b sin(t)) + 540 = 0
Now, we have a parametric equation for x and y:
x = a cos(t)
y = b sin(t)
Note that a and b can be determined based on the original hyperbola before translation.