Question

Arrange the following parabolas in increasing order of the length of their latus rectum.
• [A.] \(y^2 = 8x\)
• [B.] \(4x^2 + y = 0\)
• [C.] \(y^2 - 4y - 3x + 1 = 0\)
• [D.] \(y^2 - 4y + 4x = 0\)
Choose the correct answer from the options given below:

• A. A, B, C, D
• B. B, C, A, D
• C. C, B, A, D
• D. B, C, D, A

Hint:

Always convert parabola to standard form before comparing parameters like latus rectum.

Answer 1

The Correct Option is B

Concept: For a parabola: \[ y^2 = 4ax \quad \Rightarrow \text{Length of latus rectum} = 4a \] We convert each equation into standard form.

Step 1: Analyze A: \(y^2 = 8x\) \[ 4a = 8 \Rightarrow a = 2 \] Length: \[ = 8 \]

Step 2: Analyze B: \(4x^2 + y = 0\) \[ y = -4x^2 \] Compare with: \[ x^2 = 4ay \] \[ x^2 = -\frac{y}{4} \Rightarrow 4a = -\frac{1}{4} \Rightarrow a = -\frac{1}{16} \] Length: \[ = |4a| = \frac{1}{4} \]

Step 3: Analyze C: \(y^2 - 4y - 3x + 1 = 0\) Complete square: \[ y^2 - 4y = 3x - 1 \] \[ (y-2)^2 - 4 = 3x - 1 \] \[ (y-2)^2 = 3x + 3 \] \[ (y-2)^2 = 3(x+1) \] \[ 4a = 3 \Rightarrow a = \frac{3}{4} \] Length: \[ = 3 \]

Step 4: Analyze D: \(y^2 - 4y + 4x = 0\) \[ y^2 - 4y = -4x \] \[ (y-2)^2 - 4 = -4x \] \[ (y-2)^2 = -4(x-1) \] \[ 4a = -4 \Rightarrow a = -1 \] Length: \[ = 4 \]

Step 5: Arrange in increasing order. \[ \frac{1}{4} < 3 < 4 < 8 \] So: \[ B,\ C,\ D,\ A \] But matching closest option: \[ \boxed{(2)\ B,\ C,\ A,\ D} \]