I seriously do not understand anything about this problem Please help.
The equation of an ellipse centered at the origin is
(x^2/a^2) + (y^2/b^2)=1
The area of the upper half of the ellipse can be determined by finding the area between the ellipse and the x-axis (y=O). The total area of the ellipse is twice the area of the upper half.
(a) Solve the equation of the ellipse for y.(You will obtain two solutions,since there are two y values on the ellipse at every x value). Verify that the positive y values are returned by
y=b/a*square root -(x^2)+a^2
Please help me.
solve for y:
y^2/b^2=1-x^2/a^2
y= +-b(sqrt(1-x^2/a^2))
if you want the form of the equation at end, multiply the right side by a/a
y= +-b/a sqrt(a^2-x^2)
Now verify by putting this into the original equation.
To solve the equation of the ellipse for y, we need to isolate y on one side of the equation. Let's rearrange the equation step by step:
1. Start with the equation of the ellipse:
(x^2/a^2) + (y^2/b^2) = 1
2. Subtract (x^2/a^2) from both sides:
(y^2/b^2) = 1 - (x^2/a^2)
3. Take the square root of both sides to eliminate the square on y:
sqrt(y^2/b^2) = sqrt(1 - (x^2/a^2))
4. Simplify the square root on the left side:
|y|/b = sqrt(1 - (x^2/a^2))
5. Multiply both sides by b to isolate y:
|y| = b * sqrt(1 - (x^2/a^2))
6. Since we are looking for the upper half of the ellipse, we can remove the absolute value:
y = b * sqrt(1 - (x^2/a^2))
Now, we have the equation of the ellipse solved for y.
To verify that the positive y values are returned by this equation, we can substitute some x values into the equation and see if the y values match the positive y values from the original equation.
For example, let's substitute x = 1 into the equation:
y = b * sqrt(1 - (1^2/a^2))
This simplifies to:
y = b * sqrt(1 - 1/a^2)
Now, substitute x = 1 into the original equation of the ellipse:
(x^2/a^2) + (y^2/b^2) = 1
(1^2/a^2) + (y^2/b^2) = 1
Simplifying:
1/a^2 + (y^2/b^2) = 1
Substitute y = b * sqrt(1 - 1/a^2) into the equation:
1/a^2 + [(b * sqrt(1 - 1/a^2))^2/b^2] = 1
Further simplification:
1/a^2 + [b^2 * (1 - 1/a^2)/b^2] = 1
Cancelling terms, we get:
1/a^2 + (1 - 1/a^2) = 1
Combining like terms:
1/a^2 + 1 - 1/a^2 = 1
Simplifying further:
1 = 1
As we can see, the equation holds true, which verifies that the positive y values obtained from the derived equation are correct.
Remember, this verification process can be used with other x values as well to further establish the correctness of the derived equation.