Q83 / 170 · Manipal (MET) 2021 · Mathematics

A line cuts off on the coordinate axes positive intercepts whose sum is 4 . If it passes through $\left(\frac{9}{2},-5\right)$, its equation is

  • Correct answer
    A
    $10 x+6 y=15$
  • B
    $2 x-y=14$
  • C
    $4 x+y=13$
  • D
    None of these

Solution

Let the equation of line be $\frac{x}{a}+\frac{y}{b}=1$. Then $a+b=4$ $\begin{aligned} & \Rightarrow \quad \begin{array}{l}\Rightarrow \\ \Rightarrow \\ \therefore \quad \frac{x}{a}+\frac{y}{4-a}=1 \\ \Rightarrow \quad x(4-a)+y a\end{array} \\ & \Rightarrow a(4-a)\end{aligned}$ This passes through $\left(\frac{9}{2},-5\right)$. $\begin{array}{ll}\therefore & \frac{9}{2}(4-a)-5 a=a(4-a) \\ \Rightarrow & 36-9 a-10 a=8 a-2 a^2 \\ \Rightarrow & 2 a^2-27 a+36=0 \\ \Rightarrow & a=12 \text { or } \frac{3}{2} \text { but } a \neq 12 \\ \therefore & a=\frac{3}{2} \Rightarrow b=\frac{5}{2}\end{array}$ $\therefore$ Required equation is $\begin{array}{cc} & \frac{2 x}{3}+\frac{2 y}{5} = 1 \\ \Rightarrow \quad & 10 x+6 y=15\end{array}$

Hence, the correct answer is option: 'A'.

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