Question 1

A (3,2,0), B (5,3,2), C (-9,6,-3) are three points forming a triangle. AD , the bisector of angle BAC meets BC in D. Find the co-ordinates of D.

Accepted Answer

Since A D is the bisector of B A C. BD DC = AB AC ....(i) Now, AB = (5-3)^2+(3-2)^2+(2-0)^2 = 4+1+4 = 9 =3 A C= (-9-3)^2+(6-2)^2+(-3-0)^2 = 144+16+9 =13 BD DC = 3 13 Hence, D divides BC in the ratio 3: 13 The co-ordinates of D are ( 3(-9)+13(5) 3+13 , 3(6)+13(3) 3+13 , 3(-3)+13(2) 3+13 ) = ( 19 8 , 57 16 , 17 16 )

Question 2

Statement- 1: If the general equation x^ 2 +y^ 2 +2 x y+2 g x+2 f y+4=0 represents a pair of real lines then | g | 2. Statement- 2: The equation a x^ 2 +2 h x y+b y^ 2 +2 g x+2 f y+c=0 represents pair

Accepted Answer

The equation represents a pair of lines if 1 1 4+2 f g 1-1 f ^ 2 -1 g ^ 2 -4 1^ 2 =0 (f-g)^ 2 =0 f=g The equation becomes (x+y)^ 2 +2 g(x+y)+4=0 Which represents pair of parallel lines, which are real provided (2 ~g )^ 2 -4 4 1 0 | g | 2

Pair of Lines — Previous Year Questions & Solutions