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JEE • MATHEMATICS • PAIR OF STRAIGHT LINES

Pair of straight lines practice · Easy

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91 Easy
91 of 201
Application +4 | -1
The gradient of one of the lines given by ax2+2hxy+by2=0a x^2+2 h x y+b y^2=0 is twice that of the other, then
View answer and explanation
Correct option: C
8h2=9ab8 h^2=9 a b
Option D (9h2=8ab9 h^2=8 a b) is correct because using the given condition about the gradients and the relationships between the coefficients and the gradients leads to 9h2=8ab9h^2 = 8ab.
92 Easy
92 of 201
Analyzing +4 | -1
If θ1\theta_1 and θ2\theta_2 be the angle which the lines given by (x2+y2)(cos⁡2θsin⁡2α+sin⁡2θ)=(xtan⁡α−ysin⁡θ)2\left(x^2+y^2\right)\left(\cos ^2 \theta \sin ^2 \alpha+\sin ^2 \theta\right)=(x \tan \alpha-y \sin \theta)^2 make with the axis of xx, then for θ=π6\theta=\frac{\pi}{6}, the value of tan⁡θ1+tan⁡θ2=\tan \theta_1+\tan \theta_2=
View answer and explanation
Correct option: B
−83cosec⁡2α-\frac{8}{3} \operatorname{cosec} 2 \alpha
Option B (-83\frac{{8}}{{3}} cosec⁡2α\operatorname{cosec} 2 \alpha) is correct because it represents the sum of the tangents of the angles the lines make with the x-axis, calculated using trigonometric identities and the given equation.
93 Easy
93 of 201
Application +4 | -1
A straight line passes through the point (1,1)(1,1) and the portion of the line intercepted between xx and yy axes is divided at the point in the ratio 3:43: 4. An equation of the line is
View answer and explanation
Correct option: B
4x+3y=74 x+3 y=7
Option B (4x+3y=74 x+3 y=7) is correct because it satisfies the condition that the line passes through (1,1) and the intercepted portion is divided in the ratio 3:4.
94 Easy
94 of 201
Application +4 | -1
Reflection of the line x+y+1=0x+y+1=0 in the line lx+my+n=0l x+m y+n=0 is
View answer and explanation
Correct option: B
(x+y+1)(l2+m2)−2(l+m)(lx+my+n)=0(x+y+1)\left(l^2+m^2\right)-2(l+m)(l x+m y+n)=0
Option B ((x+y+1)(l2+m2)−2(l+m)(lx+my+n)=0(x+y+1)(l^2+m^2)-2(l+m)(l x+m y+n)=0) is correct because it represents the reflection of the line x+y+1=0 in the line lx+my+n=0.
95 Easy
95 of 201
Application +4 | -1
If one of the lines of my2+(1−m2)xy−mx2=0m y^2+\left(1-m^2\right) x y-m x^2=0 is a bisector of the angle between the lines xy=0x y=0, then mm is
View answer and explanation
Correct option: C
1
The correct answer is 1 because substituting y=xy = x into the equation and simplifying leads to m2=1m^2 = 1, and considering the geometry, m=1m = 1.
96 Easy
96 of 201
Understanding +4 | -1
The combined equation of bisectors of angles between coordinate axes, is
View answer and explanation
Correct option: B
x2−y2=0x^2-y^2=0
Option B (x2−y2=0x^2-y^2=0) is correct because the bisectors of the coordinate axes are y=x and y=-x, and their combined equation is x²-y²=0.
97 Easy
97 of 201
Analyzing +4 | -1
If one of the lines of my2+(1−m2)xy−mx2=0m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0 is a bisector of the angle between the lines xy=0xy = 0, then mm is/are
View answer and explanation
Correct option: C
±1\pm 1
Option C (±1\pm 1) is correct because when the lines y=±xy = \pm x are substituted into the equation, it leads to m=±1m = \pm 1.
98 Easy
98 of 201
Application +4 | -1
Equation of line pair perpendicular to ax2+by2+2hxy=0a x^2+b y^2+2 h x y=0
View answer and explanation
Correct option: C
bx2−2hxy+ay2=0b x^2-2 h x y+a y^2=0
Option C (bx2−2hxy+ay2=0b x^2-2 h x y+a y^2=0) is correct because it represents the equation of a line pair perpendicular to the given line pair.
99 Easy
99 of 201
Application +4 | -1
The pair of lines represented by 3ax2+5xy+(a2−2)y2=03 a x^2+5 x y+\left(a^2-2\right) y^2=0 are perpendicular to each other for
View answer and explanation
Correct option: A
Two values of aa
Option A (Two values of aa) is correct because setting the sum of the coefficients of x2x^2 and y2y^2 to zero results in a quadratic equation with two real roots for aa.
100 Easy
100 of 201
Analyzing +4 | -1
If pairs straight lines x2−2pxy−y2=0{x^2} - 2pxy - {y^2} = 0 and x2−2qxy−y2=0{x^2} - 2qxy - {y^2} = 0 be such that each pair bisects the angle between the other pair, then
View answer and explanation
Correct option: B
pq=−1pq = - 1
Option B (91 sq. units) is correct because the area of the triangle ABC, calculated using the coordinates of A, B, and C, is 91 square units.
101 Easy
101 of 201
Application +4 | -1
If the line 5x=y\sqrt{5} x=y meets the lines x=1,x=2x=1, x=2, …x=n\ldots x=n at points A1,A2,…,AnA_1, A_2, \ldots, A_n respectively, then (0A1)2+(0A2)2+…+(0An)2\left(0 A_1\right)^2+\left(0 A_2\right)^2+\ldots+\left(0 A_n\right)^2 is equal to
View answer and explanation
Correct option: B
2n3+3n2+n2 n^3+3 n^2+n
Option B (2n3+3n2+n2 n^3+3 n^2+n) is correct because it is the sum of the squares of the distances from the origin to the points A1, A2, ..., An.
102 Easy
102 of 201
Application +4 | -1
The angle between the lines in x2−xy−6y2−7x+31y−18=0{x^2} - xy - 6{y^2} - 7x + 31y - 18 = 0 is
View answer and explanation
Correct option: B
45∘45^\circ
Option B (45∘45^\circ) is correct because the angle between the lines is calculated using the formula involving the coefficients of the quadratic equation.
103 Easy
103 of 201
Application +4 | -1
If the slopes of one of the lines given by ax2+2hxy+by2=0a{x^2} + 2hxy + b{y^2} = 0 is 5 times the other, then
View answer and explanation
Correct option: A
5h2=9ab5{h^2} = 9ab
Option A (5h2=9ab5{h^2} = 9ab) is correct because it represents the condition when one slope is 5 times the other for the given equation of pair of straight lines.
104 Easy
104 of 201
Analyzing +4 | -1
If two of the straight lines represented by ax3+bx2y+cxy2+dy3=0a x^3+b x^2 y+c x y^2+d y^3=0 are at right angles, then,
View answer and explanation
Correct option: D
a2+ac+bd+d2=0a^2+a c+b d+d^2=0
Option D (a2+ac+bd+d2=0a^2+a c+b d+d^2=0) is correct because it represents the condition when two of the three lines represented by the given cubic equation are at right angles.
105 Easy
105 of 201
Application +4 | -1
The locus of the point P(h,k)\mathrm{P}(h, k), when the area of the triangle formed by the lines y=x,x+y=2y=x, x+y=2 and the line through P(h,k)P(h, k) and parallel to the xx-axis is 4h24 h^2 is
View answer and explanation
Correct option: B
2x+y−1=02 x+y-1=0
Option B (2x+y−1=02 x+y-1=0) is correct because it represents the locus of the point P(h, k) based on the given area condition.

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106 Easy
106 of 201
Application +4 | -1
If one of the lines of the pair ax2+2a x^2+2 hxy +by2=0+b y^2=0 bisects the angle between positive direction of the axes, then a, b and h satisfy the relation.
View answer and explanation
Correct option: B
a+b=−2ha+b=-2 h
Option B (a+b=−2ha+b=-2 h) is correct because substituting y=xy=x into the equation and simplifying leads to the relation a+b=−2ha+b=-2h.
107 Easy
107 of 201
Application +4 | -1
If the lines x+(a−1)y+1=0x+(a-1) y+ 1=0 and 2x+a2y−1=02 x+a^2 y-1=0 are mutually perpendicular, then .......
View answer and explanation
Correct option: D
a=−1a=-1
Option D (a=−1a=-1) is correct because it is the only real solution to the equation derived from the perpendicularity condition of the two lines.
108 Easy
108 of 201
Application +4 | -1
The equation of the bisectors of the angle between the lines represented by the equation x2−y2=0x^2-y^2=0, is
View answer and explanation
Correct option: C
xy=0x y=0
Option C (xy=0x y=0) is correct because it represents the combined equation of the bisectors of the angles between the given lines.
109 Easy
109 of 201
Application +4 | -1
The equation 4x2−24xy+11y2=04 x^2-24 x y+11 y^2=0 represents
View answer and explanation
Correct option: C
Two lines through the origin
Option C (Two lines through the origin) is correct because the given equation is a homogeneous equation of degree 2, which always represents a pair of straight lines passing through the origin.
110 Easy
110 of 201
Application +4 | -1
Let the equation of the pair of lines, y=pxy = px and y=qxy = qx can be written as (y−px)(y−qx)=0(y - px)(y - qx) = 0 Then, the equation of the pair of the angle bisectors of the line x2−4xy−5y2=0{x^2} - 4xy - 5{y^2} = 0 is
View answer and explanation
Correct option: C
x2+3xy−y2=0{x^2} + 3xy - {y^2} = 0
Option C (x2+3xy−y2=0{x^2} + 3xy - {y^2} = 0) is correct because it's the equation of the angle bisectors derived from the given pair of lines using the standard formula.
111 Easy
111 of 201
Application +4 | -1
The equation of the line parallel to lines L1≡x+2y−5=0L_1 \equiv x+2 y-5=0 and L2≡x+2y+9=0L_2 \equiv x+2 y+9=0 and dividing the distance between L1L_1 and L2L_2 in the ratio 1:61: 6 (internally), is
View answer and explanation
Correct option: A
x+2y−3=0x+2 y-3=0
Option A (x+2y−3=0x+2 y-3=0) is correct because it represents a line parallel to the given lines and divides the distance between them in the specified ratio.
112 Easy
112 of 201
Application +4 | -1
If the product of the perpendiculars drawn from the point (1,1) on the lines ax2+2hxy+by2=0a x^2+2 h x y+b y^2=0 is 1 , then
View answer and explanation
Correct option: C
h(a+b)+ab=0h(a+b)+a b=0
Option C (h(a+b)+ab=0h(a+b)+a b=0) is correct because it is derived from the condition that the product of perpendiculars from (1,1) to the lines is 1.
113 Easy
113 of 201
Application +4 | -1
Let OO be the origin, A(1,0)A(1,0) and B(0,1)B(0,1) and P(x,y)P(x, y) are points such that xy>0x y > 0 and x+y<1x+y < 1, then
View answer and explanation
Correct option: A
PP lies either inside the triangle OABO A B or in the third quadrant
The correct answer is that PP lies either inside the triangle OABOAB or in the third quadrant because xy>0xy > 0 implies PP is in the first or third quadrant, and x+y<1x + y < 1 restricts PP to below the line x+y=1x + y = 1.
114 Easy
114 of 201
Application +4 | -1
The combined equation of the two lines ax+by+ca x+b y+c =0=0 and a′x+b′y+c′=0a^{\prime} x+b^{\prime} y+c^{\prime}=0 can be written as (ax+by(a x+b y +c)(a′x+b′y+c′)=0+c)\left(a^{\prime} x+b^{\prime} y+c^{\prime}\right)=0
The equation of the angle bisectors of the lines represented by the equation 2x2+xy−3y2=02 x^{2}+x y-3 y^{2}=0 is
View answer and explanation
Correct option: D
x2−y2−10xy=0x^{2}-y^{2}-10 x y=0
Option D (x2−y2−10xy=0x^{2}-y^{2}-10 x y=0) is correct because it represents the equation of the angle bisectors derived from the given equation.
115 Easy
115 of 201
Analyzing +4 | -1
Number of points lying on the line 7x+4y+2=07 x+4 y+2=0 which is equidistant from the lines 15x2+56xy+48y2=015 x^2+56 x y+48 y^2=0 is
View answer and explanation
Correct option: C
2
Option C (2) is correct because solving the equidistant condition yields two distinct values for λ, indicating two points on the line.
116 Easy
116 of 201
Application +4 | -1
The distance between parallel lines 8x2+8xy+2y2+26x+8 x^2+8 x y+2 y^2+26 x+ 13y+15=013 y+15=0 is
View answer and explanation
Correct option: D
7510\frac{7 \sqrt{5}}{10}
Option D (7510\frac{{7 \sqrt{{5}}}}{{10}}) is correct because it is the calculated distance between the parallel lines using the given formula.
117 Easy
117 of 201
Application +4 | -1
If Lx2−10xy+12y2+5x−16y−3=0L x^2-10 x y+12 y^2+5 x-16 y-3=0 represents a pair of straight lines, find the value of LL.
View answer and explanation
Correct option: A
2
Option A (2) is correct because it is the value of L that makes the given equation represent a pair of straight lines, derived by applying the condition abc+2fgh−af2−bg2−ch2=0abc + 2fgh - af^2 - bg^2 - ch^2 = 0.
118 Easy
118 of 201
Application +4 | -1
If the equation λx2+4xy+5y2=0\lambda x^2+4 x y+5 y^2=0 represents two lines inclined at an angle π\pi, then λ=\lambda=
View answer and explanation
Correct option: B
4/54 / 5
Option B (4/54 / 5) is correct because for the lines to be inclined at an angle π\pi, they must be coincident, implying h2−ab=0h^2 - ab = 0, which leads to λ=4/5\lambda = 4/5.
119 Easy
119 of 201
Application +4 | -1
The equation to the pair of lines perpendicular to the pair of lines 3x2−4xy+y2=03 x^2-4 x y+y^2=0, is
View answer and explanation
Correct option: A
x2+4xy+3y2=0x^2+4 x y+3 y^2=0
Option A (x2+4xy+3y2=0x^2+4 x y+3 y^2=0) is correct because the equation of the perpendicular lines is obtained by interchanging the coefficients of x2x^2 and y2y^2 and changing the sign of the middle term.
120 Easy
120 of 201
Application +4 | -1
The angle between the pair of lines 2x2+5xy+2y2+3x+3  y+1=0,2{x^2} + 5xy + 2{y^2} + 3x + 3\;y + 1 = 0, is
View answer and explanation
Correct option: A
cos⁡−1(4/5){\cos ^{ - 1}}\left( {4/5} \right)
The correct answer is A (cos⁻¹(4/5)) because the angle between the pair of lines is calculated using the formula involving the coefficients of the quadratic equation.
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