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Points $\mathrm{A}$ and $\mathrm{B}$ are $\mathrm{\left ( 1,2 \right )}$ and $\mathrm{\left ( 8,9 \right )}$ . $\mathrm{A}$ point $\mathrm{P}$ lies on $\mathrm{x}$ -axis such as $\mathrm{PA + PB}$ is minimum then determine the point $\mathrm{P}$ .

Option: 1
$\mathrm{\left(\frac{15}{11}, 0\right)}$

Option: 2
$\mathrm{\left(\frac{13}{9}, 0\right)}$

Option: 3
$\mathrm{\left(\frac{16}{11}, 0\right)}$

Option: 4
$\mathrm{\left(\frac{-4}{9}, 0\right)}$

Answers (1)

best_answer

Reflection of $\mathrm{B}$ about $\mathrm{x}$ -axis is $\mathrm{B'}$
$\mathrm{\begin{aligned} & \therefore \mathrm{PB}=\mathrm{PB}^{\prime} \\ & \Rightarrow \mathrm{AP}+\mathrm{PB}\ \ \ \ \ \quad=\mathrm{AP}+\mathrm{PB}^{\prime} \end{aligned}}$
it is minimum only when $\mathrm{APB'}$ is a st. line
$\mathrm{hence \frac{2-0}{1-a}=\frac{0-(-9)}{a-8}\ \ \ \ \ \Rightarrow a=16 / 11}$

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