The consistent second-degree equation (a₁x+b₁y+c₁)(a₂x+b₂y+c₂)=k, where a₁b₁c₁a₂b₂c₂k≠0, describes a conic. Match each condition to its type.
List I
List II
(A) a₁/a₂=b₁/b₂=c₁/c₂
(P) Parabola
(B) a₁/a₂=b₁/b₂≠c₁/c₂
(Q) Hyperbola
(C) a₁/a₂≠b₁/b₂
(R) Pair of lines
(D) a₁a₂+b₁b₂=0
(S) Circle

Question

The consistent second-degree equation (a₁x+b₁y+c₁)(a₂x+b₂y+c₂)=k, where a₁b₁c₁a₂b₂c₂k≠0, describes a conic. Match each condition to its type.

List I	List II
(A) a₁/a₂=b₁/b₂=c₁/c₂	(P) Parabola
(B) a₁/a₂=b₁/b₂≠c₁/c₂	(Q) Hyperbola
(C) a₁/a₂≠b₁/b₂	(R) Pair of lines
(D) a₁a₂+b₁b₂=0	(S) Circle

TestHub · Accepted Answer

(A),(B) Parallel factors reduce the equation to a quadratic in one linear coordinate. Real consistency gives a pair of parallel lines, possibly coincident. (C) Nonparallel factors give independent linear coordinates U,V after shifting to their intersection. UV=k with k≠0 is a hyperbola. (D) The nonzero normals are perpendicular, so the factors are nonparallel and the hyperbola is rectangular. Thus A-R, B-R, C-Q, D-Q.

Mathematics - Coordinate Geometry Mixed Question with Solution | TestHub

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