First slide

Equation of line in Straight lines

Question

If $A (p, p^{2})$ falls inside the angle made by the lines $x - 2 y = 0$ and $3 x - y = 0$ , $x > 0$ then p belong to

Moderate

A

$(3, \infty)$
B

$(0, \frac{1}{2})$
C

$(\frac{1}{2}, 3)$
D

$(- 3, \frac{- 1}{2})$

Solution

If a point $A (x_{1}, y_{1})$ lies inside the angle made by the lines $L_{1} = 0, L_{2} = 0$ then $L_{A} \cdot L_{B} < 0$
Hence, $L_{A} \cdot L_{B} = (p - 2 p^{2}) (3 p - p^{2}) < 0$
Simplify
$\begin{matrix} (1 - 2 p) (3 - p) < 0 \\ (p - \frac{1}{2}) (p - 3) < 0 \end{matrix}$
Therefore, the range of p is $(\frac{1}{2}, 3)$

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