If one of the lines represented by a x 2 + 2 h x y + b y 2 = 0 passes through ( 2 , 3 ) and the other passes…

Given that: $a{x}^2+2hxy+b{y}^2=0$

Passing through $(2,3) \&\hspace{0.17em}(4,5)$

Given equation passing through $(0,0)$

As we know, Equation of line passing through two given points $y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

Equation of the line through $(0,0) \& (2,3)$: $y=\frac{3}{2}x$

$\Rightarrow 3x-2y=0$

Equation of the line through $(0,0) \& (4,5) :$ $y=\frac{5}{4}x$

$\Rightarrow 5x-4y=0$

Here, the pair of straight lines is $(3x-2y)(5x-4y)=0$

$\Rightarrow 15x^2-22xy+8y^2=0$

$\Rightarrow a=15, h=-11, b=8$

$a+2h+b=15-22+8=1$