CONTINUATION OF SAKSHI VIDYA PAGE ( ) PAIR OF STRAIGHT LINES

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Eleanore Boone
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1 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 a pair of parall lines then = g and distane etween them is * The equation ax + hxy+ y =0 and h g a f or a( a+ ) ( a+ ) y +gx +fy + =0 ( h a) form f g ax +hxy + a square if a + and (a )fg + h ( ) a rhomus if a + 0 and (a )fg + h( f g ) a retang if a + and (a )fg + h ( f g ) 0 a parallogram if a + 0 and (a )fg + h ( f g ) 0 = a and ax + hxy+ y Area is or where (x,y ) is point of intersetion h a h a * The produt of perpendiular from origin to pair of lines S is ( a ) + 4h aα + hαβ + β + gα + fβ + * From a point ( α, β ) to S is ( a ) + 4h * If ax + hxy + y e two sides of a paralgram and lx + my + n =0 is one diagonal then x y eq of the other diagonal is = l hm am hl * If the pair of lines S interset the x axis at P and Q then the ngth PQ is ald x-interept

2 x interept, PQ = g a a Similarly y interept = f * If G is the entroid of the and D is the mid point of the 3 rd side then D = 3 G Examps: (Previous EAMCET and AIEEE models) If one of the lines given y 6x xy+ 4y is 3x + 4 y, then equals : (AIEEE-004) ) ) - 3) 3 4) -3 ax + a + xy + y lie along diameters of a ir and divide the ir into four setors suh that the area of one of the setors is thrie the area of another setor then ) 3a 0a+ 3 ) 3a a+ 3 3) 3a + 0a+ 3 4) 3a + a+ 3 If the pair of lines ( ) ( ) θ+θ=π θ=π, Hints: Area of the setor r θ, 3 /4 a+ a = a+ 3 If one of the lines of my + ( m )xy mx is a isetor of the ang etween the lines xy = 0, then m is (AIEEE-007) ) ) 3) - 4) Hint : ( ) a + = 4h 4 If the slope of one of the lines is twie the slope of the other in the pair of straight lines ax + hxy + y then 8h = (EAMCET 00) ) -9a ) 9a 3) 7a 4) -7a Hint : ( ) m : m = : a p+ q = 4h pq 5 If λ x 0xy + y + 5x 6y 3=0 represents a pair of straight lines, then λ = (EAMCET 008) ) ) 3) 4) 6 The equation of the pair of isetors of the angs etween the pair of lines x axy y is x xy y Then (AIEEE 003) ) a = ) a + 3) a = 4) a + Hint: Equation to the pair of ang isetors of x axy y 0 a x y + xy ax + xy ay = is ( ) ( ) a a = = a= a= a+ = 0 7 If the pair of straight lines given y Ax Hxy By 0( H AB) with line ax + y +, then ( A 3B)( 3A B) + + = > forms an equilateral triang + + = (EAMCET 003) ) 4H ) H 3) 3H 4) Hint: Ax + Hxy + By, ax + y + form an equilateral triang ( ) ( ) Ax + Hxy + By, ax + y 3 x ay represent the same lines H ( )( ) ( )( ) A= a 3, B= 3 a, H = 4 a A+ 3B 3A+ B = 8a 8 = 64a = 4H 8 The entroid of the triang formed y the pair of straight lines x 0xy+ 7y and the line x 3y + 4 is (EAMCET 006) ) (-7/3, 7/3) ) (-8/3, 8/3) 3) (8/3, 8/3) 4) (4/3, 4/3) α, β is the entroid of the formed y the lines ax + hxy + y =0 and Hint: If ( )

3 3 α β n lx + my + n then = = l hm am hl 3( am hlm+ l ) 9 A pair of perpendiular straight lines passing through the origin and also through the point of intersetion of the urve x + y = 4 with x + y = a The set ontaining the value of a is (EAMCET 008), 3,3 4,4 5,5 ) { } ) { } 3) { } 4) { } Hint: Homogenization 0 If a + = h then the area of the triang formed y the lines ax + hxy + y =0 and x y + is (in squnits) (EAMCET 998) a a a + a + ) ) 3) 4) a + a + a a If the pair of lines joining the origin to the ommon points of x +y =4 and y=3x+ are perpendiular, if = (EAMCET 007) ) 0 ) 3 3) /5 4) 5 Hint: Homogenization * If ax +hxy + y represents i) Real and distint lines if h > a ii) Coinident lines if h = a iii) Imaginary lines if h < a * If h a then the two lines passing through origin represented y ax + y( h + h a) and ax + y ( h h a) ax + hxy + y are Sum of the slopes m + m = h, Produt of slopes mm = a and h a m m = * The ondition that slopes of pair of lines ax + hxy + y are in the ratio p : q is a( p+ q) = 4h pq * Equation of pair of straight lines passing through ( x, y ) and parall to ax + hxy + y is a( x x ) + ( y y ) + h( x x )( y y) * Equation of pair of straight lines passing through origin and perpendiular to ax + hxy+ y = 0 is x hxy+ ay * Equation of pair of lines passing through ( x, y ) and perpendiular to ax + hxy + y is ( x x ) - h( x x ) ( y y ) + a( y y ) * The produt of perpendiular from ( α, β ) to the pair of lines aα + hαβ + β ( a ) + 4h * The area of the formed y units * If θ is an aute ang etween ax + hxy + ax + hxy + ax + hxy + y, lx + my + n is y then y is n h a am hlm+ l sq

4 4 a+ h a h a Cosθ = ; Tanθ = ; Sinθ = ( a ) + 4h a + ( a ) + 4h * If ang etween pair of lines ax + hxy + y is 90 0 then a + * The equation of the pair of isetors of the angs etween ax + hxy+ y is h( x y ) = (a ) xy * The equation of the pair of isetors of the oordinate axes is x y y = ± x * If ax + hxy+ y are two sides of the and ( x, y ) is the mid point of the third side then equation of the third side is S = S axx + h( xy+ xy ) + yy = ax + hx y + y * If the pair of lines ax + hxy + y are equally inlined with the line lx + my + n then l m = a lm h * If the pair of lines ax + hxy+ y and ax + hx y + y are equally inlined then h h = a a a (or) = a h h * If the pair of lines ax + hxy + y are equally inlined with oordinate axes then h, a < 0 * ax + y + =0 and ( ax+ y) Tan α ( x ay) =0 form an isoses with equal angs as α ) If Tan α = 3 equilateral triang ) If Tan α = Right angd Isoses triang 3) If Tan α < Isoses, otuse angd triang 4) The area of the is sq units tan α ( a + ) * The equation S = ax + hxy + y + gx + fy + represent a pair of lines if i) = a + fgh af g h ii) h a, g a, f hf g gh af * The point of intersetion of the lines ax + hxy + y + gx + fy + is, a h a h * If Px (, y ) is the pt of intersetion of the line ax + hxy+ y + gx + fy + then P a h g satisfies ax + hy+ g ; hx+ y+ f ; gx+ fy+ OR = h f g f a+ h a * If θ is the ang etween the pair of lines S =0 then Cosθ = Tanθ = ( a ) + 4h a + * The pair of lines ax + hxy + y and ax + hxy + y + gx + fy+ form a parallogram if h a Equation of diagonal, AC is gx + fy + Equation of diagonal BO is xy xy where B=(x,y ) x(gh af) = (hf g) y

5 5 * If S = af ax + hxy + y + gx + fy + represent a pair of parall lines then = g and distane etween them is * The pair of lines ax + hxy + y and rhomus then (a )fg + h( f g ) * The equation ax + hxy+ g a or a( a+ ) ax + hxy + f ( a+ ) h y + gx + fy + form a ax +hxy + y +gx +fy + =0 ( h a) form f g y =0 and a square if a + and (a )fg + h ( ) a rhomus if a + 0 and (a )fg + h ( f g ) a retang if a + and (a )fg + h ( f g ) 0 a parallogram if a + 0 and (a )fg + h ( f g ) 0 ax + hxy+ y Area is or where (x,y ) is point of intersetion h a h a * The produt of perpendiular from origin to pair of lines S is ( a ) + 4h aα + hαβ + β + gα + fβ + * From a point ( α, β ) to S is ( a ) + 4h * If ( α, β ) is the entroid of the then l * The α hm β = am hl formed y the lines formed y the lines n 3( am hlm+ l ) = ax + hxy + ax + hxy + y and lx + my + n is l m i) Isoses triang if = a lm h l m ii) Right ang Isoses triang if = a and a + lm h l m iii) Rt Angd triang if a, a + lm h iv) Rt angd if al + hlm + m (Speial ase) = a and y =0 and lx + my + n

6 6 * Equation of pair of lines passing through the origin and whih are at a distane of d units from Px ( 0, y 0) is ( ) xy0 x0y = d ( x + y ) * Orthoentre of formed y lx + my + n =0 and pair of lines ax + hxy+ y =0 is H(kl, km), na ( + ) where k= am hlm+ l * The equation of pair of lines passing through ( xy ) parall to oordinate axes is ( x ) ( ) or (perpendiular to oordinate axes) x y y * If the equation ax + hxy + y represents two sides and (l, m) is orthoenter of a triang then third side is (a + ) (lx + my) = am hlm + l * If ax + hxy + y e two sides of a paralgram and lx + my + n =0 is one diagonal then x y eq of the other diagonal is = l hm am hl * If the pair of lines S interset the x axis at P and Q then the ngth PQ is ald x-interept g a f x interept, PQ = Similarly y interept = a * If f = g and fg = h then the pair of lines hxy + gx + fy + represent a square formed with the axes -If f g and fg = h then the pair of lines hxy + gx + fy + represent a retang formed with the axes fg -Area of quadrilateral = sq units h

7 -Equation of the diagonals of the retang formed y hxy + gx + fy + and axes -Equation of AC h(gx + fy) + fg Equation of OB gx fy 7 * If G is the entroid of the and D is the mid point of the 3 rd side then D = 3 G * If ax + hxy + y represent two sides of a and lx + my + n is the third side, then lous of inentre of the is the angular isetor of the pair of lines * If the line lx + my + n meets the pair of lines ax + hxy + y at A and B suh that OA = OB then lous of the mid point of AB is the angular isetor of the pair of lines Examps: (Previous EAMCET and AIEEE models) The lous represented y the equation ( x y ) ( x y ) is ) A line parall to x - axis ) a point 3) pair of lines 4) line parall to y-axis 3 If the sum of the slopes of the lines given y x xy 7y is four times their produt, then has the value (AIEEE-004) ) ) - 3) 4) - 4 If one of the lines given y 6x xy+ 4y is 3x + 4 y, then equals : (AIEEE-004) ) ) - 3) 3 4) -3 5 The equation of the image of the pair of lines y = x y the line x = is: ) y + = x ) y = x+ 3) y = x 4) y = x 6 If one of the lines given y the equation x axy 3y = oinide with one of those given y x + xy 3y and the other lines represented y them e perpendiular then ) a = -5, = ) a = 5, = - 3) a = 5, = 4) None of these ax + a + xy + y lie along diameters of a ir and divide the ir into four setors suh that the area of one of the setors is thrie the area of another setor then ) 3a 0a+ 3 ) 3a a+ 3 3) 3a + 0a+ 3 4) 3a + a+ 3 7 If the pair of lines ( )

8 8 ( ) θ+θ=π θ=π, Hints: Area of the setor r θ, 3 /4 a+ a a+ = 8 If one of the lines of my + ( m )xy mx is a isetor of the ang etween the lines xy = 0, then m is (AIEEE-007) ) ) 3) - 4) Hint : ( ) a + = 4h 9 If the slope of one of the lines is twie the slope of the other in the pair of straight lines ax + hxy + y then 8h = ) -9a ) 9a 3) 7a 4) -7a Hint : ( ) m : m = : a p+ q = 4h pq 0 The equation to the pair of isetors of the angs etween ( y mx) = ( x+ my) is ) m( x y ) + ( m) xy ) m( x y ) m xy 3) + ( ) m( x y ) m xy 4) none + ( ) The produt of the perpendiular from (-, ) to the pair of lines x 5xy+ y is (EAMCET 999) ) 4 ) 3 3) 8 4) 5/ Area of the triang formed y the lines 3x 4xy+ y, x y = 6 is (EAMCET 004) ) 6 ) 5 3) 36 4) 49 3 The area of the triang formed y the pair of straight lines ( ax+ y) - 3( x ay) and ax + y + is (EAMCET 005) ) ) 3) 4) a + ( a + ) ( a + ) 3( a + ) 4 If λ x 0xy + y + 5x 6y 3 represents a pair of straight lines, then λ = (EAMCET 008) ) ) 3) 4) 5 If the lines x + xy 35y - 4x + 44y and 5x + λ y - 8 are onurrent, then the value of λ is (EAMCET 007) ) 0 ) 3) - 4) 6 The produt of the perpendiulars distanes from the origin on the pair of straight lines x + 5xy + y + 0x + y + is (EAMCET 005) ) /5 ) /5 3) 3/5 4) 4/5 7 The ang etween the lines represented y y sin θ xysin θ + x ( os θ ) is (EAMCET 004) ) π / ) π / 6 3) π /3 4) none Sol: a + = sin θ + os θ = Ang = π / 8 If the ang θ is aute, then the aute ang etween the pair of straight lines x ( osθ sinθ) + xyosθ + y ( osθ + sinθ) is (EAMCET 000) ) θ ) θ / 3) θ /3 4) θ Sol: If α is the ang etween the lines then osα = osθ sinθ + osθ + sinθ osθ = = osθ α = θ osθ sinθ osθ sinθ + 4 os θ ( )

9 If two lines represented y ax + x y + xy + dy are mutually perpendiular, then the slope of the third line is ) a/d ) /d 3) /d 4) d/a Sol : Let m, m, m 3 e the slopes of the lines suh that mm = m m m = a/ d m = a/ d m = a/ d ( ) The equation of the pair of isetors of the angs etween the pair of lines x axy y is x xy y Then (AIEEE 003) ) a = ) a + 3) a = 4) a + Hint: Equation to the pair of ang isetors of x axy y 0 a x y + xy ax + xy ay = is ( ) ( ) a a = = a = a= a+ = 0 3 If the pair of straight lines given y Ax Hxy By 0( H AB) triang with line ax + y +, then ( A 3B)( 3A B) + + = > forms an equilateral + + = (EAMCET 003) ) 4H ) H 3) 3H 4) H Hint: Ax + Hxy + By, ax + y + form an equilateral triang ( ) ( ) Ax + Hxy + By, ax + y 3 x ay represent the same lines ( )( ) ( )( ) A= a 3, B= 3 a, H = 4 a A+ 3B 3A+ B = 8a 8 = 64a = 4H 3 The entroid of the triang formed y the pair of straight lines x 0xy+ 7y and the line x 3y + 4 is (EAMCET 006) ) (-7/3, 7/3) ) (-8/3, 8/3) 3) (8/3, 8/3) 4) (4/3, 4/3) α, β is the entroid of the formed y the lines ax + hxy + y =0 and Hint: If ( ) lx + my + n then l α hm β = am hl n 3( am hlm+ l ) = 33 If x 5xy+ y represents two sides of a triang whose entroid is (,) then the equation of third side is ) x + y + 3 ) x y 3 3) x + y 3 4) x y The orthoenter of the triang formed y the lines x + 3y and 6x + xy y is ) (, 3) ) (3, ) 3) (-, 3) 4) (/4, /4) 35 The area (in square units) of the quadrilateral formed y the two pairs of lines l x m y n( lx+ my) 0 and l x m y + n( lx my) is (EAMCET 003) n n n n ) ) 3) 4) lm lm lm 4 lm 36 Differene of slopes of the lines x (se θ - sin θ ) - tanθ xy + y sin θ is ) ) 3) 3 4) 4 Hint : put θ=π / 4 37 A pair of perpendiular straight lines passing through the origin and also through the point of intersetion of the urve x + y = 4 with x + y = a The set ontaining the value of a is (EAMCET 008), 3,3 4,4 5,5 ) { } ) { } 3) { } 4) { } 38 If a + = h then the area of the triang formed y the lines ax + hxy + y =0 and x y + is (in squnits) (EAMCET 998) a a a + a + ) ) 3) 4) a + a + a a 39 6x 5xy - 6y, 6x 5xy 6y + x + 5y form ) Square ) Rhomus 3) Retang 4) Parallogram

10 0 40 If the pair of lines joining the origin to the ommon points of x +y =4 and y=3x+ are perpendiular, if = (EAMCET 007) ) 0 ) 3 3) /5 4) 5 Hint: Homogenization 4 If x + xy+ y + 4x 8 are parall lines, then the distane etween them is (EAMCET 995) ) 4 ) 3) 4 4) 8

PAIR OF STRAIGHT LINES.

PAIR OF STRAIGHT LINES. 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