Question

The length of the latus rectum of the parabola \[{x^2} - 4x - 8y + 12 = 0\] .
A. 4
B. 6
C. 8
D. 10

Answer 1

Hint: Given is the equation calling it as parabola. We know the general equation. So we will first convert it in that form and as we call 4a as the length of the latus rectum we will compare it with the given equation to get the length. For that we will complete the perfect square by adding and subtracting 4. This is to get the equation in standard form. And then simply we will get the length!

Complete step by step solution:
Given is the equation of the parabola,
\[{x^2} - 4x - 8y + 12 = 0\]
Now we know the general equation of the parabola is \[{x^2} = 4ay\]
So we will try to get the equation above in this form.
For that we can first form a perfect square by adding 4 to the equation. And for balancing we will subtract 4.
\[{x^2} - 4x + 4 - 8y + 12 - 4 = 0\]
Now taking the perfect square and performing the subtraction of the last two terms,
\[{\left( {x - 2} \right)^2} - 8y + 8 = 0\]
transpose the terms other than the perfect square to other sides of the equation,
\[{\left( {x - 2} \right)^2} = 8y - 8\]
Now we will take 8 common and,
\[{\left( {x - 2} \right)^2} = 8\left( {y - 1} \right)\]
Now this is of the form \[{x^2} = 4ay\]
So comparing we get,
\[4a = 8\]
Thus the length of the latus rectum of the given parabola is \[4a = 8\].
Thus option 3 is correct.
So, the correct answer is “Option C”.

Note: Here note that the equation so given is of parabola but it is not in standard form. So just don’t start with solving directly. Also note that we just need to find the length of the latus rectum so don’t go the method in which you need to find the focus or any other parameters of the parabola.