First slide

Equation of line in Straight lines

Question

The point $(a^{2}, a + 1)$ lies in the angle between the lines 3x – y + 1 = 0 and x + 2y – 5 = 0 containing the origin if

Moderate

A

$a \in (- 3, 0) \cup (\frac{1}{3}, 1)$
B

$a \in (- \infty, - 3) \cup (\frac{1}{3}, 1)$
C

$a \in (- 3, \frac{1}{3})$
D

$a \in (\frac{1}{3}, \infty)$

Solution

Since origin & the point $(a^{2}, a + 1)$ lie on the some side of both the lines, therefore we have
$3 a^{2} - (a + 1) + 1 > 0 i . e . a (3 a - 1) > 0$ gives $a \in (- \infty, 0) \cup (\frac{1}{3}, \infty)$ and $a^{2} + 2 (a + 1) - 5 < 0 i . e . a^{2} + 2 a + 3 < 0 i . e . (a - 1) (a + 3) < 0 g i v e s a \in (- 3, 1)$
Intersection of the above inequalities gives $a \in (- 3, 0) \cup (\frac{1}{3}, 1)$

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