Q89 / 160 · TS EAMCET 2023 (12 May Shift 1) · Mathematics

If each of the points $(a, 4),(-2, b)$ lies on the line joining the points $(2,-1)$ and $(5,-3)$ then the point $(a, b)$ lies on the line

  • A
    $6 x+6 y-25=0$
  • B
    $x+3 y+1=0$
  • Correct answer
    C
    $2 x+6 y+1=0$
  • D
    $2 x+3 y-5=0$

Solution

Equation of line joining $(2,-1)$ and $(5,-3)$ $\begin{aligned} & y+1=\frac{-3+1}{5-2}(x-2) \\ \Rightarrow \quad & 3(y+1)=-2(x-2) \\ \Rightarrow \quad & 3 y+3=-2 x+4 \\ L: & 2 x+3 y=1 \\ \because \quad & (a, 4) \text { lies on } L, \\ & 2 a+12=1 \Rightarrow 2 a=-11 \\ \therefore \quad & a=\frac{-11}{2} \end{aligned}$ $(-2, b)$ lies on $L$ : $-4+3 b=1 \Rightarrow 3 b=5 \Rightarrow b=\frac{5}{3}$ $\therefore \quad(a, b)$ i.e. $\left(\frac{-11}{2}, \frac{5}{3}\right)$ lies on line $2 x+6 y+1=0$. $\begin{aligned} & y+1=\frac{-3+1}{5-2}(x-2) \\ \Rightarrow \quad & 3(y+1)=-2(x-2) \\ \Rightarrow \quad & 3 y+3=-2 x+4 \\ L: & 2 x+3 y=1 \\ \because \quad & (a, 4) \text { lies on } L, \\ & 2 a+12=1 \Rightarrow 2 a=-11 \\ \therefore \quad & a=\frac{-11}{2} \end{aligned}$ $(-2, b)$ lies on $L$ : $-4+3 b=1 \Rightarrow 3 b=5 \Rightarrow b=\frac{5}{3}$ $\therefore \quad(a, b)$ i.e. $\left(\frac{-11}{2}, \frac{5}{3}\right)$ lies on line $2 x+6 y+1=0$.

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

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