Prove that the lines `x = py + q; z = ry + s and x = p'y + q'; z = r'y + s'` are perpendicular If `p p' + q q' + 1 = 0`

Three lines px+qy+r=0 , qx+ry+p=0 and rx+py+q=0 are concurrent , if

The lines p(p^(2)+1)x-y+q=0 and (p^(2)+1)^(2)x+(p^(2)+1)y+2q=0 are perpendicular to a common line for

Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

If the lines p_1 x + q_1 y = 1, p_2 x + q_2 y=1 and p_3 x + q_3 y = 1 be concurrent, show that the points (p_1 , q_1), (p_2 , q_2 ) and (p_3 , q_3) are colliner.

Let image of the line (x-1)/(3)=(y-3)/(5)=(z-4)/(2) in the plane 2x-y+z+3=0 be L. A plane 7x+py+qz+r=0 is such that it contains the line L and perpendicular to the plane 2x-y+z+3=0 then p=1 (b) q=42p+q+r=30( d) p+q+r=0

P Q R S is a parallelogram. P X\ a n d\ Q Y are respectively, the perpendiculars from P\ a n d\ Q to S R\ a n d\ R S produced. Prove that P X=Q Y

SECTION MULTIPLE CORRECT CHOICE TYPE Example 41 Let a, b, c , p, q be five different non-zero real numbers and x, y, z be three numbers satisfying the system of equations a a-p aq then (a) x + y + z = a + b + c-p-q abc (b) x = pq 0 in (c) y = (a-p)(h-p)(c-p) p(p-q) (a-q)(b-q)(c-q) aq-p) (d) z = b.