If the lines `x=a_(1)y+b_(1),z=c_(1)y+d_(1)` and `x=a_(2)y+b_(2),z=c_(2)y+d_(2)` are perpendicular, prove that, `1+a_(1)a_(2)+c_(1)c_(2)=0`.

Show that the condition for the lines x= a_1z+b_1,y = c_1z+d_1 and x = a_2z+b_2, y = c_2z+ d_2 be perpendicular is a_1a_2+c_1c_2=0

The asymptotes of the hyperbola (x^(2))/(a_(1)^(2))-(y^(2))/(b_(1)^(2))=1 and (x^(2))/(a_(2)^(2))-(y^(2))/(b_(2)^(2))=1 are perpendicular to each other. Then, (a) a_(1)/a_(2)=b_(1)/b_(2) (b) a_(1)a_(2)=b_(1)b_(2) (c) a_(1)a_(2)+b_(1)b_(2)=0 (d) a_(1)-a_(2)=b_(1)-b_(2)

Show that two lines a_(1)x + b_(1) 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 (ii) Perpendicular if a_(1) a_(2) + b_(1) b_(2) = 0 .

The straight line a_(1)x+b_(1)y+c_(1)=0 anda_(2)x+b_(2)y+c_(2)=0 are parallel to each other if -

Represent the following linear equations in matrix form: a_(1)x+b_(1)y+c_(1)z+d_(1)=0 , a_(2)x+b_(2)y+c_(2)z+d_(2)=0 and a_(3)x+b_(3)y_+c_(3)z+d_(3)=0

Show that the equation of the straight line throught (alpha,beta) and through the point of intersection of the lines a_(1)x+b_(1)y+c_(1)=0 anda_(2)x+b_(2)y+c_(2)=0 is (a_(1)x+b_(1)y+c_(1))/(a_(1)alpha+b_(1)beta+c_(1))=(a_(2)x+b_(2)y+c_(2))/(a_(2)alpha+b_(2)beta+c_(2))

If the straight lines a_(1)x+b_(1)y+c=0,a_(2)x+b_(2)y+c=0anda_(3)x+b_(3)y+c=0[cne0] are concurrent , show that the points (a_(1),b_(1),(a_(2),b_(2))and(a_(3),b_(3)) are collinear.

The straight lines (x)/(a_(1))=(y)/(b_(1))=(z)/(c_(1)) and (x-2)/(a_(2))=(y-3)/(b_(2))=(z)/(c_(2)) will be parallel if -

Statement - I: Equation of bisectors of the angles between the liens x=0 and y=0 are y=+-x Statement - II : Equation of the bisectors of the angles between the lines a_(1)x+b_(1)y+c_(1)=0anda_(2)x+b_(2)y+c_(2)=0 are (a_(1)x+b_(1)y+c_(1))/(sqrt(a_(1)^(2)+b_(1)^(2)))=+-(a_(2)x+b_(2)y+c_(2))/(sqrt(a_(2)^(2)+b_(2)^(2))) (Provided a_(1)b_(2)nea_(2)b_(1)andc_(1),c_(2)gt0)

Find the coordinates of the centriod of the triangle whose vertices are ( a_(1), b_(1), c_(1)) , (a_(2), b_(2), c_(2)) and (a_(3), b_(3), c_(3)) .