The number of values of $p$ for which the lines $x + y - 1 = 0,$ $p x + 2 y + 1 = 0$ and $4 x + 2 p y + 7 = 0$ are concurrent, is

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Solution

Given that lines are concurrent.
So, $∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & - 1 p & 2 & 1 4 & 2 p & 7 \end{matrix} ∣ ∣ ∣ ∣ = 0$

$\Rightarrow - 2 p^{2} - 9 p + 26 = 0$
$\Rightarrow p = - \frac{13}{2}, 2$
But at $p = 2,$ all lines will be parallel.
$∴$ Only one value of $p$ exists.

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