First slide

Pair of straight lines

Question

If the equation of the pair of straight lines passing through the point (1,1) one making an angle $θ$ with the positive direction of the x-axis and the other making the same angle with the positive direction of the y-axis, is $x^{2} - (a + 2) xy + y^{2} + a (x + y - 1) = 0,$ $a \neq - 2,$ then the value of sin $2 θ$ is

Moderate

A

$a - 2$
B

$a + 2$
C

$2 / (a + 2)$
D

$2 / a$

Solution

The equations of the given lines are
$y - 1 = \tan θ (x - 1)$ and $y - 1 = \cot θ (x - 1)$
So, their joint equation is
$[(y - 1) - \tan θ (x - 1)] [y - 1 - \cot θ (x - 1)] = 0$
or ${(y - 1)}^{2} - (tanθ + \cot θ) (x - 1) (y - 1) + {(x - 1)}^{2} = 0$
or $x^{2} - (tanθ + \cot θ) xy + y^{2} + (\tan θ + \cot θ - 2) (x + y - 1) = 0$
Comparing with the given equation, we get $tanθ + \cot θ = a + 2$
or $\frac{1}{\sin θ cosθ} = a + 2$
or $\sin 2 θ = \frac{2}{a + 2}$

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