Show that two lines `a_(1) x + b_(14) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0`, where `b_(1) , b_(2) ne 0` are Perpendicular if `a_(1) a_(2) + b_(1) b_(2) =0`.

Show that two lines a_(1) x + b_(14) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 , where b_(1) , b_(2) ne 0 are (i) Parallel if (a_1)/( b_1) = (a_2)/( b_2) and

Statement -1 If system of equations 2x+3y=a and bx +4y=5 has infinite solution, the a=(15)/(4),b=(8)/(5) Statement-2 Straight lines a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)==0 are parallel if a_(1)/(a_(2))=b_(1)/(b_(2))nec_(1)/c_(2)

Two curves a_(i)x^(2) + b_(i)y^(2)= 1, i=1, 2 where a_(1) ne a_(2), b_(1) ne b_(2), a_(1), a_(2), b_(1), b_(2) ne 0 may intersect orthogonally If ______

Two balls , having linear momenta vec(p)_(1) = p hat(i) and vec(p)_(2) = - p hat(i) , undergo a collision in free space. There is no external force acting on the balls. Let vec(p)_(1) and vec(p)_(2) , be their final momenta. The following option(s) is (are) NOT ALLOWED for any non -zero value of p , a_(1) , a_(2) , b_(1) , b_(2) , c_(1) and c_(2) (i) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) , vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j) (ii) vec(p)_(1) = c_(1) vec(k) , vec(p)_(2) = c_(2) hat(k) (iii) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) + c_(1) hat(k) ,vec(p)_(2) = a_(2) hat(i) + b_(2) hat(j) - c_(1) hat(k) (iv) vec(p)_(1) = a_(1) hat(i) + b_(1) hat(j) , vec(p)_(2) = a_(2) hat(i) + b_(1) hat(j)

Under which condition does ther pair of linear equations a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 have infinitely many solutions?

Answer each question by selecting the proper alternative from those given below each question so as to make the statement true: The solution of the pair of linear equations a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 by corss-multiplication method is given by .................. x=(b_(2)c_(1)-b_(1)c_(2))/(a_(1)b_(2)-a_(2)b_(1)), y=(a_(1)c_(2)-a_(2)c_(1))/(a_(1)b_(2)-a_(2)b_(1)) x=(b_(1)c_(2)-b_(2)c_(1))/(a_(1)b_(2)-a_(2)b_(1)), y=(a_(1)c_(2)-a_(2)c_(1))/(a_(1)b_(2)-a_(2)b_(1)) x=(b_(2)c_(1)-b_(1)c_(2))/(a_(1)b_(2)-a_(2)b_(1)), y=(a_(2)c_(1)-a_(1)c_(2))/(a_(1)b_(2)-a_(2)b_(1)) x=(b_(1)c_(2)-b_(2)c_(1))/(a_(1)b_(2)-a_(2)b_(1)), y=(a_(2)c_(1)-a_(1)c_(2))/(a_(1)b_(2)-a_(2)b_(1))

The y-axis divides the line segment joining A(a_(1), b_(1)) and B(a_(2), b_(2)) in the ratio . . . . .

Let (a_(1),b_(1)) and (a_(2),b_(2)) are the pair of real numbers such that 10,a,b,ab constitute an arithmetic progression. Then, the value of ((2a_(1)a_(2)+b_(1)b_(2))/(10)) is

Show that the tangent of an angle between the lines (x)/(a) + (y)/( b) = 1 and (x)/( a) - (y)/( b) = 1 is (2 ab)/( a^(2) - b^(2) ) .