If The Line 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 2a+ Sin2b is Equal To       

The equation x – y = 4 and x² + 4xy + y² = 0 represent the sides of  

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 x-axis and the other making the same angle with the positive direction of y-axis is x² – (a + 2)xy + y² + a(x + y – 1) = 0,  

a ≠ –2, then the value of sin 2θ is 

If θ₁ and θ₂ be the angles which the lines (x² + y²) (cos² θ sin²α + sin² θ) = (x tan α – y sin θ)² make with axis ofx, then if θ = π/6,

tan θ₁ + tan θ₂ is equal to  

If two of the lines represented by  

                      x⁴ + x³ y + cx² y² – xy³ + y⁴ = 0  

bisect the angle between the other two, then the value of c is 

The straight line is x + y = 0, 3x + y – 4 = 0 and x + 3y – 4 = 0 from a triangle which is  

 If the line x + 2ay + a = 0, x + 3by + b = 0 and x + 4cy + c = 0 are concurrent, then a, b, c are in

If p₁, p₂ denote the lengths of the perpendiculars from the origin on the lines x sec α + y cosec α = 2a and

x cos α + y sin α = a cos 2α respectively, then  is equal to

The locus of the point of intersection of the lines x sin θ + (1 – cos θ) y = a sin θ and x sin θ – (1 + cos θ) y + a sin θ = 0 is

The straight lines 4x – 3y – 5 = 0, x – 2y – 10 = 0, 7x + 4y – 40 = 0 and x + 3y + 10 = 0 form the sides of a

If two vertices of a triangle are (5, –1) and (–2, 3), and the orthocenter lies at the origin, the coordinate of the third vertex are