If the locus of a point $P$ which is collinear with the points $A (2, 3)$ and $B (4, 7)$ is $a x + b y = 1$ , then the value of $a^{2} + b^{2}$ is

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Solution

Let $P (h, k)$ be the locus point.
$A (2, 3), B (4, 7), P (h, k)$ are collinear
$\Rightarrow$ Area of $△ A B P = 0$
$\Rightarrow \frac{1}{2} ∣ ∣ ∣ \begin{matrix} 2 & 4 & h & 2 3 & 7 & k & 3 \end{matrix} ∣ ∣ ∣ = 0$
$\Rightarrow 14 - 12 + 4 k - 7 h + 3 h - 2 k = 0 \Rightarrow - 4 h + 2 k + 2 = 0 \Rightarrow 2 h - k - 1 = 0$
Locus of $P$ is $2 x - y - 1 = 0$
$\Rightarrow 2 x - y = 1 ∴ a = 2, b = - 1 \Rightarrow a^{2} + b^{2} = 5$

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If the locus of a point P which is collinear with the points A(2,3) and B(4,7) is ax+by=1, then the value of a2+b2 is

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