In the following x,y and (a) are all variables. Show steps please! Thank you!
1) y^2 = x^2+a^2
2) y^2+ay = x^2+ax+a^2
If both are true, then ay = ax
and
therefore y = x
Use implicit diff. to find dy/dx of each of the following.
This is the question
#1: 2y y' = 2x + 0
#2: 2y y' + ay' = 2x + a + 0
just solve for y' in each case.
Sure! Let's go through the steps for each equation:
1) To simplify the equation y^2 = x^2 + a^2, you can subtract x^2 from both sides to get:
y^2 - x^2 = a^2
2) Now, let's move on to the second equation, y^2 + ay = x^2 + ax + a^2. To simplify, we can rearrange the terms:
y^2 + ay - a^2 = x^2 + ax
3) Now add a term to both sides to complete the square on the right side of the equation. To do this, take half of the coefficient of 'a' (which is 1/2) and square it (which is (1/2)^2 = 1/4). Add this term to both sides:
y^2 + ay + 1/4 - a^2 + 1/4 = x^2 + ax + 1/4
4) Simplify the left side by combining like terms and factoring the perfect square trinomial:
(y + 1/2)^2 - a^2 + 1/2 = x^2 + ax + 1/4
5) Combine the constant terms on the right side by adding 1/4 to both sides of the equation:
(y + 1/2)^2 - a^2 + 1/2 + 1/4 = x^2 + ax + 1/4 + 1/4
(y + 1/2)^2 - a^2 + 3/4 = x^2 + ax + 1/2
And these are the steps to simplify each equation.