Question

Find equation of latus rectum of the parabola \[{\left( {x + 1} \right)^2} = 32y\]?
A) \[y = 32\]
B) \[x - 8 = 0\]
C) \[y - 8 = 0\]
D) \[x = 32\]

Answer 1

Hint: First, we will use the standard equation of the parabola is \[{\left( {x - h} \right)^2} = 4a\left( {y - k} \right)\], where \[a\] is any real number and \[\left( {h,k} \right)\] is the coordinate of vertex. Then we will compare the standard equation of the parabola with the given equation to find the vertex \[\left( {h,k} \right)\]. If the value of \[a\] is positive, the parabola is an upward opening and \[a\] is negative, the parabola is a downward opening. Then we have to find the focus using the formula \[\left( {h,k + a} \right)\] for upward opening parabola and use the fact that the latus rectum passes through the focus and is perpendicular to the axis of parabola to find the required equation.

Complete step by step solution: We are given that the equation of the parabola is \[{\left( {x + 1} \right)^2} = 32y\].

We know that the standard equation of the parabola is \[{\left( {x - h} \right)^2} = 4a\left( {y - k} \right)\], where \[a\] is any real number and \[\left( {h,k} \right)\] is the coordinate of vertex.

We will now compare the standard equation of the parabola with the given equation to find the vertex \[\left( {h,k} \right)\], we get

\[ \Rightarrow \left( {h,k} \right) = \left( { - 1,0} \right)\]

\[
   \Rightarrow a = \dfrac{{32}}{4} \\
   \Rightarrow a = 8 \\
 \]

Since the value of \[a\] is positive, the parabola is an upward-opening.

We know that the vertex \[\left( {h,k} \right)\] of the standard equation of the upward opening parabola, the focus is \[\left( {h,k + a} \right)\].

Substituting the values of \[h\], \[k\] and \[a\] in the above value of focus, we get

\[
   \Rightarrow \left( { - 1,0 + 8} \right) \\
   \Rightarrow \left( { - 1,8} \right) \\
 \]

Since we also know that the latus rectum passes through the focus and is perpendicular to the axis of the parabola.

\[ \Rightarrow y = 8\]

Subtracting the above equation by 8 on each side to find the equation of the latus rectum, we get

\[
   \Rightarrow y - 8 = 8 - 8 \\
   \Rightarrow y - 8 = 0 \\
 \]

Thus, the equation of the latus rectum is \[y - 8 = 0\].

Hence, option C is correct.

Note: In solving these types of questions, you should be familiar with the concept of the standard equation of the parabola and its focus. Some students take the formula of focus \[\left( {h,k - a} \right)\] for an upward opening parabola, which is wrong. Students should use the fact of the latus rectum passes through the focus and is perpendicular to the axis of the parabola to find the required value. One should know that a latus rectum of a conic section is the chord through a focus parallel to the section and a parabola is a plane curve, which is mirror-symmetrical and is approximately U-shaped.