Question

If the line $\frac{x}{a}+\frac{y}{b}=1$ passes through the points $(2,-3)$ and $(4,-5)$, then $(a,b)=$

• A. $(1,1)$
• B. $(-1,1)$
• C. $(1,-1)$
• D. $(-1,-1)$

Answer 1

The correct option is D

$(-1,-1)$

Explanation for the correct option:

Step 1: Find the equations for $a$ and $b$.

In the question, an equation $\frac{x}{a}+\frac{y}{b}=1$ is given which passes through the points $(2,-3)$ and $(4,-5)$.

Since, the given line passes through the given points. So, the given points satisfy the given equation of line.

Substitute $x=2$ and $y=-3$ in the given equation of the line.

$$\begin{gathered} \frac{2}{a}-\frac{3}{b}=1 \\ \Rightarrow 2b-3a=ab...1 \end{gathered}$$

Now, Substitute $x=4$ and $y=-5$ in the given equation of the line.

$$\begin{gathered} \frac{4}{a}-\frac{5}{b}=1 \\ \Rightarrow 4b-5a=ab...2 \end{gathered}$$

Step 2: Find the value of $a$ and $b$.

Multiply $2$ on both sides of equation $1$ and subtract from equation $2$.

$$\begin{gathered} 4b-6a-4b+5a=2ab-ab \\ \Rightarrow -a=ab \\ \Rightarrow b=-1 \end{gathered}$$

Multiply $5$ on both sides of equation $1$, multiply $3$ on both sides of equation $2$ and subtract.

$$\begin{gathered} 10b-15a-12b+15a=5ab-3ab \\ \Rightarrow -2b=2ab \\ \Rightarrow a=-1 \end{gathered}$$

Therefore, the value of $\left(a,b\right)$ is $(-1,-1)$.

Hence, option (D) is the correct answer.