A differential equation representing the family of parabolas with...

The equation of family of parabolas

$( x - \alpha{)}^{2} = 4a ( y - \beta )$
Distance of point $( 2, - 3 )$ from line $3x + 4y = 5$ is
$4a = \left|\dfrac{3 \times 2 + 4 \times \left(- 3\right) - 5}{5}\right|$

$4a = \left|\dfrac{6 - 12 - 5}{5}\right|$

$4a = \dfrac{11}{5}$

$( x - \alpha{)}^{2} = \dfrac{11}{5} ( y - \beta )$
Differentiate w.r.t. $x$
$2 ( x - \alpha ) = \dfrac{11}{5}\dfrac{dy}{dx}$

Again diff. w. r. t. $x$

$2 = \dfrac{11}{5}\dfrac{{d}^{2}y}{d{x}^{2}}$

$11\dfrac{{d}^{2}y}{d{x}^{2}} = 10$