Class 11 Mathematics Straight Lines MCQs Set 10

Question. If the pair of lines \( 2x^2 + 3xy + y^2 = 0 \) makes angles \( \theta_1 \) and \( \theta_2 \) with X-axis then \( \tan(\theta_1 - \theta_2) = \)
(a) 1
(b) 1/2
(c) 1/3
(d) 1/4
Answer: (c) 1/3

Question. The triangle formed by \( x + 3y = 1 \) and \( 9x^2 - 12xy + ky^2 = 0 \) is right angled triangle and \( k \neq -9 \). Then k =
(a) 3
(b) 5
(c) 7
(d) 1
Answer: (a) 3

Question. If the equation \( 2x^2 - 5xy + 2y^2 = 0 \) represents two sides of an isosceles triangle then the equation of the third side passing through the point (3,3) is
(a) \( x+y = 3 \)
(b) \( x-y=0 \)
(c) \( 2x-y=3 \)
(d) \( x+y-6=0 \)
Answer: (d) x+y-6=0

Question. If \( px^2 - y^2 + 3x + 11y + q = 0 \) represents a pair of perpendicular lines then (p, q) =
(a) (–1, –14)
(b) (–1, 28)
(c) (1, –28)
(d) (–1, –28)
Answer: (d) (–1, –28)

Question. The pair of lines represented by \( 3x^2 + 5xy + (a^2 - 2)y^2 = 0 \) are perpendicular to each other for
(a) Two values of a
(b) for all values of a
(c) for one value of a
(d) for no value of a
Answer: (a) Two values of a

Question. If the equation \( x^2 + py^2 + y = a^2 \) represents a pair of perpendicular lines, then the point of intersection of the lines is
(a) (1, a)
(b) (1, -a)
(c) (0, a)
(d) (0, 2a)
Answer: (c) (0, a)

Question. If the pair of lines \( 3x^2 - 5xy + py^2 = 0 \) and \( 6x^2 - xy - 5y^2 = 0 \) have one line in common, then p =
(a) \( 2, \frac{25}{4} \)
(b) \( -2, \frac{25}{4} \)
(c) \( -2, \frac{-25}{4} \)
(d) \( 2, \frac{-25}{4} \)
Answer: (d) \( 2, \frac{-25}{4} \)

Question. If one of the lines represented by \( 3x^2 - 4xy + y^2 = 0 \) is perpendicular to one of the line \( 2x^2 - 5xy + ky^2 = 0 \) then k =
(a) -3, 13/9
(b) 3, -13/9
(c) -3, -13/9
(d) -7, -33
Answer: (d) -7, -33

Question. The condition that one of the pair of lines \( ax^2 + 2hxy + by^2 = 0 \) be coincident with one line of the pair \( 3x^2 + 12xy + 2y^2 = 0 \) and the remaining lines are at right angles, then h (a-b) =
(a) a + b
(b) ab
(c) 2 ab
(d) a / b
Answer: (b) ab

Question. The line \( y = 3x \) bisects the angle between the lines \( ax^2 + 2axy + y^2 = 0 \) if a =
(a) 3
(b) 11
(c) 3/11
(d) 11/3
Answer: (c) 3/11

Question. The centroid of the triangle formed by the pair of straight lines \( 12x^2 - 20xy + 7y^2 = 0 \) and the line \( 2x - 3y + 4 = 0 \) is
(a) \( \left( \frac{7}{3}, \frac{7}{3} \right) \)
(b) \( \left( \frac{8}{3}, \frac{8}{3} \right) \)
(c) \( \left( \frac{8}{3}, \frac{8}{3} \right) \)
(d) \( \left( \frac{4}{3}, \frac{4}{3} \right) \)
Answer: (c) \( \left( \frac{8}{3}, \frac{8}{3} \right) \)

Question. If \( 2x^2 - 5xy + 2y^2 = 0 \) represents two sides of a triangle whose centroid is (1, 1) then the equation of the third side is
(a) x+y-3=0
(b) x-y-3=0
(c) x+y+3=0
(d) x-y+3=0
Answer: (a) x+y-3=0

Question. The orthocentre of the triangle formed by the lines \( x^2 - 3y^2 = 0 \) and the line \( x = a \) is
(a) \( \left( \frac{a}{3}, 0 \right) \)
(b) \( \left( \frac{2a}{3}, 0 \right) \)
(c) (a, 0)
(d) \( \left( \frac{4a}{3}, 0 \right) \)
Answer: (b) \( \left( \frac{2a}{3}, 0 \right) \)

Question. If \( x^2 + 4xy + y^2 = 0 \) represents two sides of \( \Delta OAB \) and the orthocentre is (-1, -1), then the third side is
(a) x+y = 2
(b) x+y=1
(c) x+y+1=0
(d) x+y=3
Answer: (b) x+y=1

Question. The circumcentre of the triangle formed by the lines \( x^2 - y^2 = 0 \) and \( y - 5 = 0 \) is
(a) (5, 0)
(b) (0, 5)
(c) (0, 0)
(d) (5, 5)
Answer: (b) (0, 5)

Question. The orthocentre of the triangle formed by the lines \( x + 3y - 10 = 0 \) and \( 6x^2 + xy - y^2 = 0 \) is
(a) (1, 3)
(b) (3, 1)
(c) (-1, 3)
(d) (1, -3)
Answer: (a) (1, 3)

Question. The distance from the point of intersection of the lines \( x^2 - y^2 - 2x + 2y = 0 \) to the point of intersection of the lines \( 2x^2 - 5xy + 2y^2 + x + y - 1 = 0 \) is
(a) 1
(b) 0
(c) 2
(d) 4
Answer: (b) 0

Question. If \( \frac{x}{a} + \frac{y}{b} = 1 \) intersects \( 5x^2 + 5y^2 + 5bx + 5ay - 9ab = 0 \) at P and Q, \( \angle POQ = \pi / 2 \) then the relation between a and b is
(a) a = b
(b) a = 2b or b = 2a
(c) a = 3b or b = 3a
(d) a + b = 5
Answer: (b) a = 2b or b = 2a

Question. If the pair of lines which joins the origin to the point of intersection of \( ax^2 + 2hxy + by^2 + 2gx = 0 \), \( a_1x^2 + 2h_1xy + b_1y^2 + 2g_1x = 0 \) are at right angles then
(a) \( \frac{g}{g_1} = \frac{a+b}{a_1+b_1} \)
(b) \( \frac{g}{g_1} = \frac{a+b}{a_1+b_1} \)
(c) \( \frac{h}{h_1} = \frac{a+b}{a_1+b_1} \)
(d) \( \frac{h}{h_1} = \frac{a_1+b_1}{a+b} \)
Answer: (b) \( \frac{g}{g_1} = \frac{a+b}{a_1+b_1} \)

Question. The angle between the lines joining the origin to the point of intersection of \( lx + my = 1 \) and \( x^2 + y^2 = a^2 \) is
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{4} \)
(c) \( \cos^{-1} \left( \frac{1}{a\sqrt{l^2+m^2}} \right) \)
(d) \( 2\cos^{-1} \left( \frac{1}{a\sqrt{l^2+m^2}} \right) \)
Answer: (d) \( 2\cos^{-1} \left( \frac{1}{a\sqrt{l^2+m^2}} \right) \)

Question. The combined equation of the pair of lines passing through origin which are at a distance 4 units from the point (5, 6) is
(a) \( 9x^2 + 60xy - 20y^2 = 0 \)
(b) \( 9x^2 - 60xy + 20y^2 = 0 \)
(c) \( 20x^2 + 60xy - 9y^2 = 0 \)
(d) \( 20x^2 - 60xy + 9y^2 = 0 \)
Answer: (d) \( 20x^2 - 60xy + 9y^2 = 0 \)

Question. Perpendiculars AL, AM are drawn from any point A on the x-axis to the pair of lines \( 2x^2 - 5xy - 3y^2 = 0 \) the angle made by LM with +ve direction of x-axis is
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{4} \)
(d) \( \frac{\pi}{2} \)
Answer: (c) \( \frac{\pi}{4} \)

Question. Two of the lines represented by \( ax^3 + 3bx^2y + 3cxy^2 + dy^3 = 0 \) will be perpendicular if
(a) \( a^2 + ac + db + d^2 = 0 \)
(b) \( a^2 + 3(ac + bd) + d^2 = 0 \)
(c) \( a^2 - 3(ac + bd) + d^2 = 0 \)
(d) \( a^2 - ac - bd + d^2 = 0 \)
Answer: (b) \( a^2 + 3(ac + bd) + d^2 = 0 \)

Question. The line \( x + y = 1 \) meets the lines represented by the equation \( y^3 - xy^2 - 14x^2y + 24x^3 = 0 \) at the points A, B, C. If O is the point of intersection of the lines represented by the given equation then \( OA^2 + OB^2 + OC^2 = \)
(a) 22/9
(b) 85/72
(c) 181/72
(d) 221/72
Answer: (d) 221/72

Question. If two lines represented by \( x^4 + x^3y + cx^2y^2 - xy^3 + y^4 = 0 \) bisect the angle between the other two, then the value of ‘c’ is
(a) 0
(b) -1
(c) 1
(d) -6
Answer: (d) -6

Question. If a and b positive numbers (\( a < b \)), then the range of values of k for which a real \( \lambda \) be found such that equation \( ax^2 + 2\lambda xy + by^2 + 2k(x+y+1) = 0 \) represents a pair of straight lines is
(a) \( a < k^2 < b \)
(b) \( a \le k^2 \le b \)
(c) \( k^2 \le a \) or \( k^2 \ge b \)
(d) \( k \le a^2 \) or \( k \ge 2b \)
Answer: (d) \( k \le a^2 \) or \( k \ge 2b \)

Question. The triangle formed by the pair of lines \( x^2 - 4y^2 = 0 \) and the line \( x - a = 0 \) is always
(a) Equilateral
(b) Isosceles
(c) Right angled
(d) Scalene
Answer: (b) Isosceles

Question. The lines \( 33y^2 - 136xy + 135x^2 = 0 \) are equally inclined to
(a) x+2y + 7 = 0
(b) 2x + y – 7 = 0
(c) x + 2y – 7 = 0
(d) x + y = 1
Answer: (b) 2x + y – 7 = 0

Question. If the equation \( 6x^2 + 5xy + by^2 + 9x + 20y + c = 0 \) represents a pair of perpendicular lines, then b - c =
(a) – 6
(b) – 3
(c) – 2
(d) 0
Answer: (d) 0

Question. If \( ax^2 + 6xy + by^2 - 10x + 10y - 6 = 0 \) represents a pair of perpendicular straight lines, then |a| is equal to
(a) 2
(b) 4
(c) 1
(d) 3
Answer: (b) 4

Question. If the equation \( \lambda x^2 - 5xy + 6y^2 + x - 3y = 0 \) represents a pair of straight lines then their point of intersection is
(a) (-3, -1)
(b) (-1, -3)
(c) (3, 1)
(d) (1, 3)
Answer: (a) (-3, -1)

Question. If the two pairs of lines \( 3x^2 - 5xy + 2y^2 = 0 \) and \( 6x^2 - xy + ky^2 = 0 \) have one line in common, then \( k^2 + 7k - 10 = \)
(a) 0
(b) -20
(c) -1
(d) 2
Answer: (b) -20

Question. If one of the lines represented by \( 2x^2 + 2hxy + 3y^2 = 0 \) be perpendicular to one of the lines given by \( 3x^2 + 2h_1xy + 2y^2 = 0 \), then
(a) \( h - h_1 = 0 \)
(b) \( h + h_1 = 0 \)
(c) \( 3h = 2h_1 \)
(d) \( 2h = 3h_1 \)
Answer: (b) \( h + h_1 = 0 \)

Question. If one of the lines given by \( 6x^2 - xy + 4cy^2 = 0 \) is \( 3x + 4y = 0 \), then c equals
(a) 1
(b) -1
(c) 3
(d) -3
Answer: (d) -3

Question. Assertion (A) : If two sides of a triangle are represented by \( x^2 - 3xy + 2y^2 = 0 \) and centroid is \( \left( \frac{2}{3}, 0 \right) \), then the third side is \( 2x - 3y - 2 = 0 \).
Reason (R): If two sides of a triangle are represented by \( ax^2 + 2hxy + by^2 = 0 \) and centroid is \( \left( \frac{2x_1}{3}, \frac{2y_1}{3} \right) \) then the third side is \( x(ax_1 + hy_1) + y(hx_1 + by_1) = ax_1^2 + 2hx_1y_1 + by_1^2 \)
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true and R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is the correct explanation of A

Question. The orthocentre of the triangle formed by the lines \( 2x+y=2 \) and \( 2x^2 + 3xy - 2y^2 = 0 \) is
(a) (4/3, -2/3)
(b) ( 1/2 ,1)
(c) (0,0)
(d) (1, 1)
Answer: (c) (0,0)

Question. Assertion A: If two sides of a triangle represented by \( 2x^2 + 4xy - y^2 = 0 \) and orthocentre is (1,1) then the third side is \( x + y + 3 = 0 \).
Reason R: If two sides of a triangle represented by \( ax^2 + 2hxy + by^2 = 0 \) and orthocentre is (c, d) then the third side is \( (a + b)(cx + dy) = ad^2 - 2hcd + bc^2 \)
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true and R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
Answer: (a) Both A and R are true and R is the correct explanation of A

Question. The circumcentre of the triangle formed by the lines \( 2x^2 - 3xy - 2y^2 = 0 \) and \( 3x - y = 10 \) is
(a) (2, 1)
(b) (1, -2)
(c) (3, -6)
(d) (3, -1)
Answer: (d) (3, -1)

Question. The coordinates of the orthocentre of the triangle formed by the lines \( 2x^2 - 3xy + y^2 = 0 \) and x+y=1 are is
(a) (1, 1)
(b) (1/2, 1/2)
(c) (1/3, 1/3)
(d) (1/4, 1/4)
Answer: (b) (1/2, 1/2)

Question. The area of the triangle formed by the two rays whose combined equation is \( y = |x| \) and the line \( x + 2y = 2 \).
(a) 4sq units
(b) 4/3 sq units
(c) 8/3 sq units
(d) 16/3 sq units
Answer: (b) 4/3 sq units

Question. The curve \( x^2 + y^2 + 2gx + 2fy + c = 0 \) intercepts on the line \( lx + my = 1 \), a length which subtends a right angle at the origin, then \( \frac{lg + mf + 1}{l^2 + m^2} = \)
(a) c/2
(b) -c/2
(c) 2/c
(d) -2/c
Answer: (b) -c/2

Question. The lines joining the origin to the points of intersection of \( x^2 + y^2 + 2gx + c = 0 \) and \( x^2 + y^2 + 2fy - c = 0 \) are at right angles is
(a) \( g^2 + f^2 = c \)
(b) \( g^2 - f^2 = 0 \)
(c) \( g^2 - f^2 = 2c \)
(d) \( g^2 + f^2 = c^2 \)
Answer: (c) \( g^2 - f^2 = 2c \)

Question. The line \( 4y - 3x + 48 = 0 \) cuts the curve \( y^2 = 64x \) in A and B. If AB subtends an angle \( \theta \) at the origin, then \( \tan\theta = \)
(a) 20/9
(b) 10/9
(c) 5/9
(d) 40/9
Answer: (a) 20/9

Question. If the equation \( ax^3 + 3bx^2y + 3cxy^2 + dy^3 = 0 \) (\( a,b,c,d \neq 0 \)) represents three coincident lines, then
(a) a = c
(b) b = d
(c) \( \frac{a}{b} = \frac{b}{c} = \frac{c}{d} \)
(d) ac = bd
Answer: (c) \( \frac{a}{b} = \frac{b}{c} = \frac{c}{d} \)

Question. If the line \( y = \sqrt{3}x \) cuts the curve \( x^3 + y^3 + 3xy + 5x^2 + 3y^2 + 4x + 5y - 1 = 0 \) at the points A,B,C, then \( OA.OB.OC \) is
(a) \( \frac{4}{13}(3\sqrt{3}-1) \)
(b) \( 3\sqrt{3}+1 \)
(c) \( \frac{2}{\sqrt{3}} + 7 \)
(d) \( \frac{4}{13}(3\sqrt{3}+1) \)
Answer: (a) \( \frac{4}{13}(3\sqrt{3}-1) \)

Question. The equation \( a(x^4 + y^4) - 4bxy(x^2 - y^2) + 6cx^2y^2 = 0 \) represents two pairs of lines at right angles. The two pairs will coincide if
(a) \( b^2 = a + 3c \)
(b) \( a^2 = b^2 - 3ac \)
(c) \( a^2 + b^2 = 3ac \)
(d) \( 2b^2 = a^2 + 3ac \)
Answer: (d) \( 2b^2 = a^2 + 3ac \)