ellipse equations

Question

I do not know how to change this equation: 16= 16x^2+y^2 into a standard ellipse equation

Answer 1

The standard equation of an ellipse is (x^2/b^2) + (y^2/a^2) = 1

In order to change the equation given into this form, we have to make the equation equal to 1 by dividing both sides of the equation by 16.

1) 16/16 = 16x^2/16 + y^2/16

The left side of the equation, 16/16, simplifies to be 1

The right side of the equation, 16x^2/16 + y^2/16, simplifies to be x^2/1 + y^2/16

2) 1 = x^2/1 + y^2/16

Now, the standard equation of an ellipse says (x^2/b^2) + (y^2/a^2) = 1

In the standard equation of an ellipse, the denominator of x^2 is b^2

In our equation, x^2/1 + y^2/16 = 1, the denominator of x^2 is 1.

Therefore, b^2 = 1

If we square root both sides, we find that b = 1

If we substitute 1 for b, in the equation, x^2/b^2 + y^2 = 1, we get x^2/1^2 + y^2/16 = 1

In the standard equation of an ellipse, x^2/b^2 + y^2/a^2 = 1, the denominator of y^2 is a^2

In our equation, x^2/1 + y^2/16 = 1, the denominator of y^2 is 16

Therefore, a^2 = 16

If we square root both sides, a = 4

If we substitute 4 for a in the equation, x^2/b^2 + y^2/a^2 = 1, we get x^2/1^2 + y^2/4^2 = 1, which is the final answer.

Hope this helped! Please let me know if you have any other questions! :)