Question

X^2 +4x= - (y^2 -6)

write the conic equation in standard form

Answer 1

Explanation:

`{x}^(2) + 4x = - {y}^(2) + 6`

`{x}^(2) + 4x + {y}^(2) - 6 = 0`

Standard Form of Conic is

`ax {}^(2) + bxy + c{y}^(2) + dx + ey + f = 0`

In this scenario, a is 1. c is 1 d is 4 and f is -6

Since a=c, we will get a circle.

Standard form of a circle is

`(x - h) {}^(2) + (y - k) {}^(2) = {r}^(2)`

where (h,k) is the center and r is the radius.

Step 1: Convert x^2+4x+y^2-6 into circle.

`{x}^(2) + 4x + {y}^(2) - 6`

Add 6 to both sides

`{x}^(2) + 4x + {y}^(2) = 6`

Complete the square with respect to x variable

Divide 4 by 2 and square it, then add that to both sides

`( (4)/(2) ) {}^(2) = 4`

So add four to both sides

`{x}^(2) + 4x + 4 + {y}^(2) = 6 + 4`

Remebrr that a perfect square is

`{a}^(2) + 2ab + {b}^(2) = (a + b) {}^(2)`

So we get

`(x + 2) {}^(2) + {y}^(2) = 10`

That is our standard equation.

Look at the graph above.