Question:
The order of the differential equation representing the family of all parabolas having a fixed latus rectum of length \(4a\) and an axis parallel to the x-axis is...
The order of the differential equation representing the family of all parabolas having a fixed latus rectum of length \(4a\) and an axis parallel to the x-axis is...
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The order equals the number of independent arbitrary constants in the family.
Updated On: Nov 12, 2026
- \(1\)
- \(2\)
- \(3\)
- \(4\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
The order of the differential equation of a family equals the number of independent arbitrary constants in its equation.
Step 2: Write the family
Parabolas with axis parallel to the x-axis and latus rectum \(4a\) (fixed) have equation \((y-k)^{2}=\pm4a(x-h)\).
Step 3: Count the constants
Here \(a\) is fixed, so only \(h\) and \(k\) are arbitrary: 2 constants.
Step 4: Conclude
Two constants need two differentiations to eliminate, so the order is 2.
Final Answer:
The order of the differential equation is 2. \[ \boxed{\text{(B) }2} \]
The order of the differential equation of a family equals the number of independent arbitrary constants in its equation.
Step 2: Write the family
Parabolas with axis parallel to the x-axis and latus rectum \(4a\) (fixed) have equation \((y-k)^{2}=\pm4a(x-h)\).
Step 3: Count the constants
Here \(a\) is fixed, so only \(h\) and \(k\) are arbitrary: 2 constants.
Step 4: Conclude
Two constants need two differentiations to eliminate, so the order is 2.
Final Answer:
The order of the differential equation is 2. \[ \boxed{\text{(B) }2} \]
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