Content text 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 < abk2 < 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