The correct answer is 11 d²x/dy²=10. Hence the correct option is a.
1. Form of Equation:
Since the parabolas have their axis parallel to the y-axis, their equation can be written in the form:
x^2 = 4ay + C
where "a" is a constant related to the parabola's shape and "C" is a constant related to its vertical position.
2. Latus Rectum and Constant Coefficient:
The length of the latus rectum (4a) is related to the distance of the point (2, -3) from the line 3x + 4y = 5 through the following formula:
4a = |(2)(3) + (-3)(4) - 5| / sqrt(3^2 + 4^2) = 11 / sqrt(25)
Simplifying this equation gives:
a = 11 / 10
This means the constant coefficient "C" in the parabola equation depends on "a":
C = (2^2) * (4a) - 3 = 176 / 10
3. Differential Equation:
Differentiating the parabola equation twice with respect to y and considering the above values for "a" and "C":
d²x/dy² = 16a = 10d²x/dy² = 10
Therefore, the correct differential equation for this specific family of parabolas is:
10d²x/dy² = 10
4. Matching Options:
Only option A. 11 d²x/dy²=10 includes both the correct coefficient for the second derivative (10) and the latus rectum-related factor (11). The other options either miss the latus rectum factor (B and D) or have an incorrect overall coefficient (C).
The differential equation A. 11 d²x/dy²=10 accurately represents the family of parabolas with the specified characteristics due to its consideration of both the latus rectum length and the distance of the point (2, -3) from the line. Hence the correct option is a.