If the lines 2( sin a+ sin b) x-2 sin (a-b) y=3 and 2( cos a + cos b) x+2 cos (a-b) y=5 are perpendicular, then sin 2 a+ sin 2 b is equal to

Question

If the lines 2( sin a+ sin b) x-2 sin (a-b) y=3 and 2( cos a + cos b) x+2 cos (a-b) y=5 are perpendicular, then sin 2 a+ sin 2 b is equal to (A)  sin (a-b)-2 sin (a+b) (B)  sin 2(a-b)-2 sin (a+b) (C) 2 sin (a-b)- sin (a+b) (D)  sin 2(a-b)- sin (a+b)

Tardigrade Admin · Accepted Answer

We have ( sin a+ sin b/ sin (a-b)) ×-( cos a+ cos b/ cos (a-b))=-1 ⇒ sin 2 a+ sin 2 b+2 sin (a+b)= sin 2(a-b) ∴ sin 2 a+ sin 2 b= sin 2(a-b)-2 sin (a+b)

Q. If the lines $2 (sin a + sin b) x - 2 sin (a - b) y = 3$ and $2 (cos a$ $+ cos b) x + 2 cos (a - b) y = 5$ are perpendicular, then $sin 2 a +$ $sin 2 b$ is equal to

$sin (a - b) - 2 sin (a + b)$

$sin 2 (a - b) - 2 sin (a + b)$

$2 sin (a - b) - sin (a + b)$

$sin 2 (a - b) - sin (a + b)$

We have $\frac{s i n a + s i n b}{s i n ( a - b )} \times - \frac{c o s a + c o s b}{c o s ( a - b )} = - 1$
$\Rightarrow sin 2 a + sin 2 b + 2 sin (a + b) = sin 2 (a - b)$
$∴ sin 2 a + sin 2 b = sin 2 (a - b) - 2 sin (a + b)$

Q. If the lines 2(sina+sinb)x−2sin(a−b)y=3 and 2(cosa +cosb)x+2cos(a−b)y=5 are perpendicular, then sin2a+ sin2b is equal to

sin(a−b)−2sin(a+b)

sin2(a−b)−2sin(a+b)

2sin(a−b)−sin(a+b)

sin2(a−b)−sin(a+b)