The coordinates of the focus of a parabola can be determined from its equation. Generally all equations of parabolas can be converted into one of its standard forms.
Since we already know the foci of the standard forms, it becomes quite easy to find the focus from the equations reduced to match the standard forms.
Let us take some examples:
Comparing the coefficients of 'x' in both we get
Putting `a = 1/2` in it, we get
Comparing the coefficients of 'x' in both we get
Putting `a = 2` in it, we get
The steps to convert it to a standard form are given below:
First, convert the equation to match a standard form by the method of completing the square. This works as follows:
Factor out coefficient of x squared
Since we already know the foci of the standard forms, it becomes quite easy to find the focus from the equations reduced to match the standard forms.
Let us take some examples:
Example 1: `y^2 = 2x`
This equation is already in the standard form `y^2 = 4ax` (If you haven't read about standard forms then read it here).Comparing the coefficients of 'x' in both we get
`4a = 2`so
`a = 1/2`We already know that the focus of the standard parabola `y^2 = 4ax` is `(a, 0)`. (If you don't know how we got that, see this post.)
Putting `a = 1/2` in it, we get
`(1/2, 0)`Thus, the focus of the parabola `y^2 = 2x` is `(1/2, 0)`.
Example 2: `x^2 = -8y`
Again this equation is already in the standard form `x^2 = -4ay` (If you haven't read about standard forms then read it here).Comparing the coefficients of 'x' in both we get
`4a = 8`so
`a = 2`We already know that the focus of the standard parabola `x^2 = -4ay` is `(0, -a)`. (If you don't know how we got that, see this post.)
Putting `a = 2` in it, we get
`(2, 0)`Thus, the focus of the parabola `x^2 = -8y` is `(0, -2)`.
Example 3: `y = 2x^2 + 3x + 1`
This equation does not match with any of the four standard forms. We are going to follow some simple steps to convert it to standard form. These steps are discussed in detail in this post:The steps to convert it to a standard form are given below:
First, convert the equation to match a standard form by the method of completing the square. This works as follows:
Factor out coefficient of x squared
`y = 2(x^2 + 3/2x + 1)`Take the coefficient of 'x', divide it by 2, square it and add and subtract it from the equation
`y = 2(x^2 + 3/2x + (3/4)^2 - (3/4)^2 + 1)`Take the part containing x squared, x, and the squared term you added above and compare it with the formula `a^2 + 2ab + b^2 = (a + b)^2`. Then you get
`y = 2((x + 3/4)^2 - (3/4)^2 + 1)`Simplify the numerical part:
`y = 2((x + 3/4)^2 - 13/4)`Move the constants over to the right side:
`y/2 = (x + 3/4)^2 - 13/4`
`y/2 + 13/4 = (x + 3/4)^2`Again, factor out the coefficient of y from the right hand side,
`1/2(y + 13/2) = (x + 3/4)^2`Or
`(x + 3/4)^2 = 1/2(y + 13/2)`Now we can change this equation to resemble the standard equation `x^2 = 4ay`. In order to do so, we will replace x + 3/4 with X and y + 13/2 with Y. That is,
`X^2 = 1/2Y`
...where `X = x + 3/4` and `Y = y + 13/2`Note: If you are having trouble with what we did above then read this:
Compare the above equation with the standard form `x^2 = 4ay`. Thus we have
`4a = 1/2`
`a = 1/8`We already know that the focus of the parabola `x^2 = 4ay` lies on (0, a). Hence, the focus of the parabola above lies on (0, 1/8). But here we have taken
`X = x + 3/4`Which means that
`x = X - 3/4`And
`Y = y + 13/2`Which means that
`y = Y - 13/2`That is, the actual values of x and y coordinates of any point are `(X - 3/4, Y - 13/2)`. Thus, since the focus is (0, 1/8) hence the coordinates of the actual focus are
`(-3/4, 1/8 - 13/2)`That is,
`(-3/4, -51/8)`Hence the coordinates of the focus of the parabola `y = 2x^2 + 3x + 1` are `(-3/4, -51/8)`.
