Question

A differential equation representing the family of parabolas with axis parallel to $y-$axis and whose length of latus rectum is the distance of the point $(2,-3)$ from the line $3x+4y=5,$ is given by

• A. $11\frac{d^{2}y}{d{x}^2}=10$
• B. $11\frac{d^{2}x}{d{y}^2}=10$
• C. $10\frac{d^{2}x}{d{y}^2}=11$
• D. $10\frac{d^{2}y}{d{x}^2}=11$

Answer 1

The correct option is A $11\frac{d^{2}y}{d{x}^2}=10$

Length of latus rectum $=\frac{|3\times 2+4\times (-3)-5|}{\sqrt{3^2+4^2}}=\frac{11}{5}$
Let equation of parabola : $(x-a{)}^2=\frac{11}{5}(y-b)$
$\Rightarrow 2(x-a)=\frac{11}{5}\frac{dy}{dx}$
$\Rightarrow 2=\frac{11}{5}\frac{d^{2}y}{d{x}^2}$
$\Rightarrow 11\frac{d^{2}y}{d{x}^2}=10$