PAIR OF STRAIGHT LINES.

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1 PAIR OF STRAIGHT LINES PREVIOUS EAMCET BITS 1. The value of λ with λ < 16 suh that x 1xy + 1y + 5x + λy 3 = represets a pair of straight lies, is [EAMCET 9] 1) 1 ) 9 3) 1 4)9 As: Sol. Δ= λ= 9. The area (i square uits) of the triagle formed by x + y + 1 = ad the pair of straight lies x 3xy + y = is [EAMCET 9] 1) 7/1) 5/1 3) 1/1 4) 1/6 Sol. Area = h ab 1 = am h m + b 1 3. The pairs of straight lies x 3xy + y = ad x 3xy + y + x = form a [EAMCET 9] 1) square but ot rhombus ) rhombus 3) parallelogram 4) retagle but ot a square Sol. I give two equatios homogeious parts are same, hee is a parallelogram. 4. The value of λ suh that λx 1xy+ 1y + 5x 16y 3= represets a pair of straight lies, is [EAMCET 8] 1) 1 ) 1 3) 4) Sol. λx 1xy + 1y + 5x 16y 3 = ax + hxy + by + gx + fy + = a =λ,b= 1,= 3,f = 8,g= 5/,h = 5 ab + fgh af bg h = ( )( ) ( )( )( ) ( ) ( ) ( )( ) λ / 5 λ 8 1 5/ 3 5 = 36λ+ 64λ = 1λ= λ= 5. A pair of perpediular straight lies passes through the origi ad also through the poit of itersetio of the urve x + y = 4 with x + y = a. The set otaiig the value of a is [EAMCET 8] 1) {, } ) { 3,3} 3) { 4, 4} 4) { 5,5} As: 1 Sol. Clearly the poit of itersetio of x + y = 4,x+ y= ais the origi 1

2 ( ) ( ) x + y = 4, x + y = a x + a x = 4, x = a = 4 a = ± Pair of straight lies 6. The agle betwee the pair of straight lies formed by joiig the poits of itersetio of x + y = 4 ad y= 3x+ to the origi is a right agle. The = [EAMCET 7] 1) ) 13 3) 1/5 4) 5 As: 1 Sol. = a ( 1+ m ) = 4 = 7. If the lies x + xy 35y 4x+ 44y 1 = ad 5x + λy 8 = are ourret, the the value of λ is [EAMCET 7] 1) ) 1 3) 1 4) 4 Sol. Poit of itersetio of give pair of lies is, 3 3 It lies o 5x +λy 8 = λ= 8. The lies represeted by the equatio x y x 3y + = are [EAMCET 6] 1) x+ y 1=,x y+ = ) x y =,x+ y+ 1= 3) x+ y+ =,x y 1= 4) x y+ 1=,x+ y = Sol. By verifiatio ( x y+ 1)( x+ y ) = x + y x+ 3y = 9. The etroid of the triagle formed by the pair of straight lies 1x xy + 7y = ad the lie x 3y + 4 = is [EAMCET 6] 7 7 1), ), ), 3 3 Sol. Cetroid divides media i the ratio of : 1 3x1 3y1 of ( x,y 1 1) is etroid the, must lie o 8 8 x 3y + 4 = by verifiatio, is etroid ), 3 3 O(,) 1. The area of the triagle formed by the pair of straight lies (ax + by) 3(bx ay) = ad ax + by + = 1 is [EAMCET 5] 1

3 Pair of straight lies 1) a + b ) ( + b) a 3) ( + ) a b 4) ( + ) 3 a b Sol. Y lar distae from (, ) to ax + by + = is h = Area a + b h = 3 3 a b ( + ) O (,) X 11. The produt of the perpediular distaes from the origi o the pair of straight lies 1x + 5xy + 1y + 1x + 11y + = is [EAMCET 5] 1) 1 5 As: ) 5 Sol. Produt of perpediulars ( a b) + ( h) 3) 3 5 = = 5 4) The agle betwee the lies represeted by ( ) 1) 3 Sol. a+ b= θ= 9 ) Area of the triagle formed by the lies y si θ xysi θ+ x os θ 1 = is 3) 6 4) [EAMCET 4] 3x 4xy + y =,x y = 6is [EAMCET 4] 1) 16 ) 5 3) 36 4) 49 h ab Sol. Area of the triagle = = 36 sq.uits am h m + b + + = > forms a equilateral triagle with lie ax + by + =, the (A + 3B) (3A + B) =.. [EAMCET 3] 1) H ) H 3) H 4) 4H 14. If the pair of straight lies give by Ax Hxy By O( H AB) 3

4 Pair of straight lies Sol. Equatio of oe side ax + by + = ad other two sides passig through origi the the equatio ax + by 3 bx ay = of other two sides is ( ) ( ) A = a 3b ; B = b 3a ; H = 8ab ( A+ 3B)( 3A+ B) = 4H 15. The area (i square uits) of the quadrilateral formed by the two pairs of lies ( ) ( ) x m y x+ my = ad x m y + zx my = [EAMCET 3] 1).m ).m As: 1 x m y x+ my = Sol. ( ) ( )( ) ( ) ( )( ) x+ my x my = x m y + x my = x my x+ my+ = The give lies from a rhombus Area of rhombus = = ab m 3).m 4) 4.m 16. If the oordiate axes are the bisetors of the agle betwee the pair of lies ax +hxy+b =, where h > ab ad a b, the [EAMCET ] 1) a+ b= ) h = 3) h,a+ b= 4) a+ b As: h x y a b xy= Sol. Equatio of pair of agular bisetors is ( ) ( ) Equatio of oordiate axes is xy = h = 17. If the agle θ is aute the the aute agle betwee the pair of straight lies x (osθ - siθ) + xy osθ + y (osθ + siθ) = [EAMCET ] 1) θ ) θ/ 3) θ/3 4) θ Sol. os α= a+ b a b + 4h ( ) (α is agle betwee the pair of lies ad a = osθ + siθ, b = osθ- siθ h = osθ) osα = osθ Agle θ 4

5 5 Pair of straight lies 18. If the pair of straight lies xy x y + 1 = ad the lie ax + y 3 = are ourret, the a = [EAMCET ] 1) ) 3 3) 1 4) Sol. xy x y + 1 = ( x 1)( y 1) = ( x, y) = ( 1,1) ax + y 3 = lie passes through (1, 1) a = The orthoetre of the triagle formed by the lies x + 3y = 1 ad 6x + xy y = is [EAMCET 1] 1) (1, 3) ) (3, 1) 3) ( 1, 3) 4) (1, 3) As: 1 Sol. The give lies are x + 3y = 1, 6x + xy y = 1 = 1, m = 3, = 1, a = 6, h = ; b = 1 ( a+ b) Orthoetre = (( k;km ) where k = am h m + b. If oe of the lies of the pair of straight lies ax + hxy + by = bisets the agle betwee the o-ordiate axes the : [EAMCET 1] 1) a b h + = ) ( ) a+ b = h 3) a b 4h Sol. The agle bisetors of the axes are x± y= ax ± hx + bx = ( ) a+ b = 4h + = 4) ( ) a+ b = 4h 1. If the slope of oe lie is twie the slope of the other i the pair of straight lies ax + hxy + by =, the 8h = [EAMCET 1] 1) 9ab ) 9ab 3) 7ab 4) 7ab As: Sol. If slope of oe lie is k times to other the 4kh = k + 1 ab 8h = 9ab ( ). The equatio of the pair of lies through the poit (a, b) parallel to the oordiate axes is [EAMCET ] 1) ( x b)( y a) = ) ( )( ) 3)( x a)( y b) = 4) ( )( ) Sol. The equatio of the oordiate axes is xy = x b y+ b = x+ a y b =

6 Pair of straight lies The equatio of the pair of lies passig through (a, b) ad parallel to xy = is (x a) (y b) = 6

CONTINUATION OF SAKSHI VIDYA PAGE ( ) PAIR OF STRAIGHT LINES

CONTINUATION OF SAKSHI VIDYA PAGE (0--008) A Bhanu Kumar, Senior Faulty, Sri Chaitanya Eduational Institutions, Hyderaad PAIR OF STRAIGHT LINES (I Year Inter) * If S = af ax + hxy + y + gx + fy + represent