a. An ellipse is formed by stretching the graph of x^2+ y^2=1 horizontally by a factor of 3 and vertically by a factor of 4. Determine the equation of the ellipse in standard form

x^2/a^2+y^2/b^2=1
(x/3) ^2+ (y/4) ^2=1
x^2/9 + y^2/16 =1

so the equation is:
x^2/9 + y^2/16 =1

b. The ellipse in question above is translated 4 units to the right and 2 units down. Determine the equations of this ellipse. Express your answer in both standard form and general form:

so the equation for standard form is:
(x-h)^2/a^2 + (y-k)^2/b^2=1

so
a: 9
b: 16
x: ?
Y?
h=4
k: 2

(x-4) ^2/9^2 + (y-2)^2/16^2=1

is this standard form right or am I wrong?

Yes, you are correct. The standard form equation of the translated ellipse is:

(x-4)^2/9^2 + (y-2)^2/16^2 = 1

In this equation, the center of the ellipse is shifted 4 units to the right (h=4) and 2 units down (k=2) from the origin. The major axis (2a) of the ellipse is 2 times the square root of 9, which is equal to 6, and the minor axis (2b) is 2 times the square root of 16, which is equal to 8.

Therefore, plugging in these values into the standard form equation of an ellipse, we get:

(x-4)^2/6^2 + (y-2)^2/8^2 = 1