Tiêu đề XI - maths - chapter 10 - PAIR OF STRAIGHT LINES FINAL (92-117).pdf ✅

PAIR OF STRAIGHT LINES JEE-MAIN-JR-MATHS VOL-III 92 NARAYANAGROUP Homogeneous equations Combined Equation of a Pair of Straight lines :  i)If 1 2 L L   0, 0 are any two lines, then the combined equation of 1 2 L L   0, 0 is 1 2 L L  0 ii) Any second degree equation in x and y represents a pair of straight lines if the expression on the left hand side can be expressed as a product of two linear factors in x and y. Separate equations of pair of lines :  The equations of the separate lines of 2 2 ax hxy by    2 0are   2 ax h h ab y    0,   2 ax h h ab y     0 Nature of pair of lines :  The second degree homogeneous equation 2 2 ax hxy by    2 0 represents a pair of straight lines passing through the origin and it represents (i) two real and distinct lines if 2 h ab  (ii) two coincident lines if 2 h ab  (iii) Imaginary lines if 2 h ab  W.E-1:- The equation 2x2 +kxy+2y2 = 0 represents a pair of imaginary lines if Sol :The given equation will represent a pair of imaginary lines if h2 < abk2 < 16 (k - 4) (k + 4) < 0  k   4,4 Slopes of pair of lines :  i) If 1 2 y m x y m x   , are the two lines represented by the pair of lines 2 2 ax h xy b y    2 0 , b  0 with slopes m1 and m2 then a) The slopes of the lines are the roots of the quadratic equation 2 bm hm a    2 0 b) 2 1 2 1 2 1 2 2 2 ; ; h a h ab m m m m m m b b b        c) The combined equation of pair of lines with slopes m1 , m2 is  y m x y m x    1 2   0   2 2 1 2 1 2      y m m xy m m x 0 ii) The slopes of the straight lines represented by 2 2 ax hxy by    2 0 are reciprocal to each other if a b  iii) If the slopes of two lines represented by 2 2 ax hxy by    2 0 are in the ratio l m: then   2 2 l m ab h lm   4 iv) If the slope of one of the lines represented by 2 2 ax hxy by    2 0 is k times the slope of other line then   2 2 4 1 kh k ab   v) 2 2 ax hxy by    2 0 represents a pair of lines if the slope of one line is the nth power of the other then     1/ 1 1/ 1 2 0 n n n n ab a b h      vi) If the slope of one line of pair of lines 2 2 ax hxy by    2 0is square of the slope of the other line then   3 ab a b h h     6 8 0 Angle between the pair of lines :  If  is an acute angle between the pair of lines 2 2 ax hxy by    2 0 then   2 2 cos 4 a b a b h      or   2 2 2 2 sin 4 h ab a b h      or 2 2 tan h ab a b     ; a b   0 i)The lines represented by 2 2 ax hxy by    2 0 are perpendicular , if a b   0. i.e., coefficient of 2 x  coefficient of 2 y  0 SYNOPSIS PAIR OF STRAIGHT LINES
JEE-MAIN-JR-MATHS VOL-III PAIR OF STRAIGHT LINES NARAYANAGROUP 93 Types of triangles :  i) The equation of the pair of lines passing through the origin and forming an isosceles triangle with the line ax by c    0 is     2 2 ax by k bx ay     0 . (a) If k 1 then the triangle is right angled isosceles. (b) If k  3 then the triangle is equilateral. (c) If 1 3 k  then the triangle is an isosceles and obtuse angled ii) The triangle formed by the pair of lines 2 2 S ax hxy by     2 0 and the line lx my n    0 is a) equilateral if 2 2 ax hxy by    2     2 2 lx my mx ly    3 b) Isosceles if     2 2 h l m a b lm    c) Right angled if a b   0 or S l m  , 0   W.E-2:- The triangle formed by the lines 2 2 2 3 2 0,3 1 0 x xy y x y       is Sol: Given line is 3x+y+1 = 0 The simplification of (3x+y)2 - (x-3y)2 =0 is given pair of lines. It is in the form (ax + by)2 - k(bx - ay)2 = 0 Here k = 1  The triangle is right angled isosceles. Centres related with triangles :  i) If  ,  is the centroid of the triangle whose sides are 2 2 ax hxy by    2 0 and lx my n    0 , then   2 2 2 3 2 n bl hm am hl bl hlm am          (or)      ,  , , 3 l m l m l m n F F F x y                      where 2 2 F bx hxy ay    2 Pair of parallel & perpendicular lines :  i) The equation to the pair of lines passing through the point  x y 1 1 ,  and parallel to the pair of straight lines 2 2 ax hxy by    2 0 is        2 2 1 1 1 1 a x x h x x y y b y y        2 0 ii) The equation to the pair of lines passing through the origin and perpendicular to 2 2 ax hxy by    2 0 is 2 2 bx hxy ay    2 0 iii) The equation to the pair of lines passing through the point  x y 1 1 ,  and perpendicular to the pair of straight lines 2 2 ax hxy by    2 0 is        2 2 b x x h x x y y a y y        1 1 1 1 2 0 Common line to pair of lines :  i) If the pairs of lines 2 2 1 1 1 a x h xy b y    2 0 , 2 2 2 2 2 a x h xy b y    2 0 have one line in common then 2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 . 2 2  a h h b a b a h h b a b (or)      2 1 2 2 1 1 2 2 1 1 2 2 1 a b a b h a h a h b h b      4 0 ii) If one of the lines represented by 2 2 1 1 1 a x h xy b y    2 0 is perpendicular to one of the lines represented by 2 2 2 2 2 a x h xy b y    2 0 then 2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 . 2 2    a h h b a b b h h a b a (or)      2 1 2 1 2 1 2 2 1 1 2 2 1 a a bb h a h b hb h a      4 0 iii) If the pair of lines 2 2 1 1 1 a x h xy b y    2 0 and 2 2 2 2 2 a x h xy b y    2 0 are such that they have one line in common and the remaining lines are perpendicular then 1 2 1 1 2 2 1 1 1 1 h h a b a b               
PAIR OF STRAIGHT LINES JEE-MAIN-JR-MATHS VOL-III 94 NARAYANAGROUP ii) The pair of lines 2 2 S ax hxy by     2 0 represents two sides of a triangle and  x y 1 1 ,  is the mid point of the third side then the equation of third side is 1 11 S S  i.e.,   2 2 1 1 1 1 1 1 1 1 axx h xy x y byy ax hx y by       2 iii) If 2 2 ax hxy by    2 0 represents two sides of a triangle, G x y  1 1 ,  be its centroid then the mid point of the third side of the triangle is 3 3 3 1 1 . ., , 2 2 2 x y G i e       iv) If kl km,  is the orthocentre of the triangle formed by the lines 2 2 ax hxy by    2 0 and lx my n    0 then   2 2 2 n a b k am hlm bl      v) The distance from the origin to the orthocentre of the triangle formed by the lines 1 x y     and 2 2 ax hxy by    2 0 is   2 2 2 2 2 a b a h b           vi) If 2 2 ax hxy by    2 0 represents two sides of a triangle for which c d,  is the orthocentre, then the equation of the third side of triangle is    2 2 a b cx dy ad hcd bc      2 Product of perpendiculars :  i) The product of the perpendiculars from  ,  to the pair of lines 2 2 ax hxy by    2 0 is   2 2 2 2 2 4 a h b a b h        Area of the triangle :  i) The area of the triangle formed by the line lx my n    0 and the pair of lines 2 2 ax hxy by    2 0 is 2 2 2 2 2 n h ab am hlm bl    ii) The equation of the pair of lines through the origin and making an angle ' '  with the line lx my n    0 is     2 2 2 lx my mx ly     tan 0  and the area of the triangle is   2 2 2 tan n  l m iii) The area of an equilateral triangle formed by the line ax by c    0 with the pair of lines     2 2 ax by bx ay     3 0 is   2 2 2 3 c a b  2 3 p  where p is the perpendicular distance from the origin to the line ax by c    0 W.E-3:- If     2 2 2 3 36 3 2 0 x y x y     and 2 3 4 5 0 x y    represents an isosceles triangle with base angle 1 tan 6  then its area is Sol :Equation of given line is 2 3 4 5 0 x y    here l  2 , m  3 , n  4 5 Given that tan 6    Area of the triangle =   2 2 2 tan n  l m = 16 5 8 6 5 3    sq.units Pair of angular bisectors :  i) The equation to the pair of bisectors of the angles between the pair of straight lines 2 2 ax hxy by    2 0 is     2 2 h x y a b xy    ii) The angle between pair of angular bisectors of any pair of lines is 2  . iii) The equation to the pair of bisectors of the coordinate axes is 2 2 x y   0 iv) If one of the line in 2 2 ax hxy by    2 0 bisects the angle between the coordinate axes then   2 2 a b h   4
JEE-MAIN-JR-MATHS VOL-III PAIR OF STRAIGHT LINES NARAYANAGROUP 95 W.E-4:- Equation of the bisectors of the angles between the lines through the origin and the sum and product of whose slopes are respectively the arithmetic and geometric means of 9 and 16 is Sol: A.M. of 9 and 16 = 1 2 9 16 25 2 2 m m     G..M. of 9 and 16 = 1 2 9 16 12 .    m m Equation of pair of lines is   2 2 1 2 1 2 y m m xy m m x     0 2 2     24 25 2 0 x xy y Equation of the bisectors is     2 2 h x y a b xy     2 2 25 44 25 0 x xy y    Equally inclined with a line :  i) A pair of lines 1 2 L L  0 is said to be equally inclined to a line L  0 if the lines 1 2 L L   0, 0 subtend the same angle with the line L  0 ii) Every pair of lines is equally inclined to either of its angular bisectors iii) A pair of lines is equally inclined to a line L  0, if L  0 is parallel to one of the angular bisectors. iv)Given pair of lines through origin is equally inclined to the coordinate axes  the pair of angular bisectors of given pair of lines through origin is the coordinate axes v) If the pair of lines 2 2 ax hxy by    2 0 equally inclined to the coordinate axes then h  0 and ab  0 vi) The pair of lines 1 2 L L  0 bisects the angle between the pair of lines 3 4 L L  0 pair of angular bisectors of 3 4 L L  0 and pair of lines 1 2 L L  0 represents the same equation vii) Two pairs of lines 1 2 L L  0, 3 4 L L  0 are such that each bisects the angle between the other pair  pair of angular bisector of 1 2 L L  0 , pair of lines 3 4 L L  0 represents same and vice versa. viii) Two pairs of lines are equally inclined to each other  two pairs of lines have same pair of angular bisectors W.E-5:- The lines ax2 + 2hxy + by2 = 0 are equally inclined to the lines ax2 + 2hxy + by2 +  (x2 + y2 ) = 0 for what values of  ? Sol: Equation of the bisectors of the angle between the lines ax2 + 2hxy + by2 +  (x2 + y2 ) = 0 is h(x2 - y2 ) = (a - b)xy. Which is same as the equation of the bisectors of angles between the lines ax2 + 2hxy + by2 = 0 The given two pairs of lines are equally inclined to each other for any value of  . Non homogeneous equations: Condition for pair of lines :  i) If the equation 2 2 S ax hxy by gx fy c        2 2 2 0 represents a pair of lines then a) 2 2 2        abc fgh af bg ch 2 0 i.e 0 a h g h b f g f c  b) 2 2 2 h ab g ac f bc    , , W.E-6:-If   2 2 ax by fy c a      2 0, 0 represents a pair of lines then f is Sol :   2 2 ax by fy c a      2 0, 0 represents a pair of lines then it satisfy the condition   0 i.e. 2 abc af   0 2   f bc  f is G..M. between b and c. ii) If 2 2 ax hxy by gx fy c       2 2 2 0 represents a pair of lines then 2 2 ax hxy by    2 0 represents a pair of lines parallel to them and passing through the origin Angle between the pair of lines :  i) The angle between the pair of lines 2 2 ax hxy by gx fy c       2 2 2 0 is same as the angle between the pair of lines 2 2 ax hxy by    2 0