$\begin {array} {1 1} (A)\;(1, 1) & \quad (B)\;(-1,1) \\ (C)\;(1,-1) & \quad (D)\;(-1,-1) \end {array}$

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- If a line $ax+by+c=0$ passes through a point $(x_1, y_1)$, then the coordinates can be substituted in the place of $x$ and $y$.

Step 1:

Given equation of the line is

$ \large\frac{x}{a}$$+\large\frac{y}{b}$$=1$

$ \Rightarrow bx+ay=ab$

It is given that this line passes through (2, -3)

$ \therefore b(2)+a(-3)=ab$

(i.e) $2b-3a=ab$--------(1)

It also passes through (4, -5)

$ \therefore 4b-5a=ab$--------(2)

On solving equation (1) and (2)

$ (\times 2 ) 2b-3a=ab$

$\qquad 4b-5a=ab$

__________________

$\qquad 4b-6a=2ab$

$\qquad 4b-5a=ab$

$ \quad (-) \quad (+) \quad (-)$

_________________

$\quad \qquad -a = ab$

$ \therefore b = -1$

and substituting for $b$ in equation (1) we get,

$2(-1)-3a=ab$

$-2-3a=a(-1)$

$ -3a+a=2$

$-2a=2\qquad a=-1$$

Hence $a=-1$ and $b = -1$

Hence D is the correct option.

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