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Solved Examples: How to find the length of latus rectum of a parabola?

Example 1: `x^2 = 8y`

on comparing it with `x^2 = 4ay` , we get
`4a = 8`
`a = 2`
and thus length of latus rectum is
Latus Rectum `= 4a = 4 * 2 = 8`
Quite simply, the length of the latus rectum is equal the the coefficient of the y variable here.

But the coefficient can be negative, as in `x^2 = -8y`. The length of the latus rectum remains positive in both cases. That is, it is equal to 8 in both the equations above.

Now suppose you have the equation

Example 2: `y^2 = 12x`

This is comparable with the standard equation
`y^2 = 4ax`
Thus we get
`4a = 12`
`a = 3`
Hence length of latus rectum is
Latus Rectum `= 4 * a = 4 * 3 = 12`
Now suppose we have an equation which is not in the standard form. Let us take, for example

Example 3: `y = 2x^2 + 3x + 1` 

Method 1

In order to find the length of the latus rectum for this parabolic equation, we will first convert it into the standard form. (To do that, you need to follow some steps which are discussed in detail here: Change Parabola to Standard Form). It proceeds as follows:
`y = 2(x^2 + 3/2x + 1/2)`
`y = 2(x + 3/2x + (3/4)^2- (3/4)^2 + 1/2)`
The highlighted part above is compared with the formula
`a^2 + 2ab + b^2 = (a + b)^2`
Thus we can write
`y = 2((x + 3/4)^2 - 9/14 + 1/2)`
Simplifying,
`y = 2((x + 3/4)^2 - 1/7)`
Since we notice that the `x` containing term is squared, not `y`, so we know that this equation will be compared to the standard form `x^2 = +/- 4ay`. Hence we will remove all numbers being multiplied to or on the side of the `x` squared term.

Dividing by 2,
`y/2 = (x + 3/4)^2 - 1/7`
Adding 1/7 to both sides,
`y/2 + 1/7 = (x + 3/4)^2`

Method 2

`(x + 3/4)^2 = y/2 + 1/7`
Now we factor out the coefficient of `y`
`(x + 3/4)^2 = 1/2(y + 2/7)`
To make it look more like `x^2 = 4ay`, we will replace `x + 3/4` with `X` and `y + 2/7` with `Y`. Thus we get,
`X^2 = 1/2Y`, where `X = x + 3/4` and `Y = y + 2/7`
This equation of the parabola is same as the original one above, that is, `y = 2x^2 + 3x + 1` and in a different form comparable with the standard form `x^2 = 4ay`.

Now comparing it with `x^2 = 4ay`, we get
`4a = 1/2`
`a = 1/8`
Thus the length of the latus rectum is
Latus Rectum `= 4*a = 4*1/8 = 1/2`
Thus, to find the length of the latus rectum of a parabola you first need to convert the equation into one of its standard forms, and then find the value of `4a` by comparing it with the standard forms.

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