question_answer

If in a \[\Delta ABC\], \[DE||BC\]and intersects AB at D and AC at E, then match the lists

	Column - I		Column - II
P.	\[\frac{AD}{DB}\]	1.	\[\frac{AC}{AE}\]
Q.	\[\frac{AB}{AD}\]	2.	\[\frac{AE}{EC}\]
R.	\[\frac{DB}{AB}\]	3.	\[\frac{AB}{AC}\]
S.	\[\frac{AD}{AE}\]	4.	\[\frac{EC}{AC}\]

A) P-1,      Q-2,      R-3,      S-4

B) P-4,      Q-3,      R-2,      S-1

C) P-2,      Q-1,      R-4,      S-3

D) P-1,      Q-3,     R-2,      S-4

Correct Answer: C

Solution :

In  \[\Delta \,ABC,\]\[BC\,\,|\,\,|\,\,DE\]

(P)  \[\frac{AD}{DB}=\frac{AE}{EC}\left( BPT \right)\]

(Q) \[\frac{AD}{DB}=\frac{AE}{EC}\]

\[\Rightarrow \frac{DB}{AD}=\frac{EC}{AE}\Rightarrow \frac{DB}{AD}+1=\frac{EC}{AE}+1\]

\[\Rightarrow \frac{BD+AD}{AD}=\frac{EC+AE}{AE}\Rightarrow \frac{AB}{AD}=\frac{AC}{AE}\]

(R) Similarly, \[\frac{DB}{AB}=\frac{AB}{AC}\]

(S) \[\frac{AD}{AB}=\frac{AE}{AC}\Rightarrow \frac{AD}{AE}=\frac{AB}{AC}\]