Explanation

(b)
Let $P(4,\ 1)$and $PD\bot AB$.
Equation of $AB$ is $3x - 2y - 8 = 0$
$\therefore$ Equation of $PD$ is $2x + 3y - 11 = 0$
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Let line $AB$ is divided by $PD$ in the ratio λ:1, then intersecting point
$D\left( \frac{6\lambda + 2}{\lambda + 1},\frac{5\lambda - 1}{\lambda + 1} \right)\ lies\ on\ 2x + 3y - 11 = 0$
$\Rightarrow \ 2\left( \frac{6\lambda + 2}{\lambda + 1} \right) + 3\left( \frac{5\lambda - 1}{\lambda + 1} \right) - 11 = 0$
$\Rightarrow \ 16\lambda - 10 = 0$
$\Rightarrow \ \ \lambda:1 = 5:8$