Class 11 Mathematics Straight Lines MCQs Set 27

Question. The equation of a straight line L is \( x + y = 2 \), and \( L_1 \) is another straight line perpendicular to L and passes through the point (1/2, 0), then area of the triangle formed by the y-axis and the lines L, \( L_1 \) is
(a) 25/8
(b) 25/16
(c) 25/4
(d) 25/12
Answer: (b) 25/16

Question. In an isosceles triangle OAB, O is the origin and OA = OB = 6. The equation of the side AB is \( x - y + 1 = 0 \). Then the area of the triangle is
(a) \( 2\sqrt{21} \)
(b) \( \sqrt{142} \)
(c) \( \frac{\sqrt{142}}{2} \)
(d) \( \frac{\sqrt{71}}{2} \)
Answer: (d) \( \frac{\sqrt{71}}{2} \)

Question. An equilateral triangle is constructed between two parallel lines \( \sqrt{3}x + y - 6 = 0 \) and \( \sqrt{3}x + y + 9 = 0 \) with base on one and vertex on the other. Then the area of triangle is
(a) \( \frac{200}{\sqrt{3}} \)
(b) \( \frac{225}{4\sqrt{3}} \)
(c) \( \frac{225}{\sqrt{3}} \)
(d) \( \frac{200}{4\sqrt{3}} \)
Answer: (b) \( \frac{225}{4\sqrt{3}} \)

Question. Area of triangle formed by the lines \( 2x + y - 3 = 0 \), \( x + 4y - 5 = 0 \) and \( 3x + 5y - 1 = 0 \) is
(a) 15/2
(b) 49/2
(c) 27/56
(d) 7/2
Answer: (d) 7/2

Question. If \( f(x + y) = f(x)f(y) \) for all x and y if \( f(1) = 2 \), then area enclosed by \( 3|x| + 2|y| \leq 8 \) is
(a) f(5) sq.units
(b) f(6) sq.units
(c) 1/3 f(6) sq.units
(d) f(4) sq.units
Answer: (c) 1/3 f(6) sq.units

Question. Four sides of a quadrilateral are given by the equation \( xy(x-2)(y-3) = 0 \), then the equation of the line parallel to \( x - 4y = 0 \) that divides the quadrilateral into two equal parts is
(a) \( x - 4y + 5 = 0 \)
(b) \( x - 4y - 5 = 0 \)
(c) \( x - 4y + 1 = 0 \)
(d) \( x - 4y - 1 = 0 \)
Answer: (a) \( x - 4y + 5 = 0 \)

Question. \( L_1 \) and \( L_2 \) are two intersecting lines and the angle between the image of \( L_1 \) w.r.t \( L_2 \) and that of \( L_2 \) w.r.t \( L_1 \) is \( 45^\circ \). Then the angle between \( L_1 \) and \( L_2 \) is
(a) \( 20^\circ \)
(b) \( 15^\circ \)
(c) \( 45^\circ \)
(d) \( 60^\circ \)
Answer: (b) \( 15^\circ \)

Question. \( L_1 \) and \( L_2 \) are two intersecting lines. If the image of \( L_1 \) w.r.t. \( L_2 \) and that of \( L_2 \) w.r.t. \( L_1 \) coincide, then the angle between \( L_1 \) and \( L_2 \) is
(a) \( 35^\circ \)
(b) \( 60^\circ \)
(c) \( 90^\circ \)
(d) \( 45^\circ \)
Answer: (b) \( 60^\circ \)

Question. For all values of \( \theta \) all the lines represented by the equation \( (2\cos\theta + 3\sin\theta)x + (3\cos\theta - 5\sin\theta)y - (5\cos\theta - 2\sin\theta) = 0 \) passes through a fixed point then the reflection of that point with respect to the line \( x+y = \sqrt{2} \) is
(a) \( (\sqrt{2}+1, \sqrt{2}+1) \)
(b) \( (\sqrt{2}-1, \sqrt{2}-1) \)
(c) \( (\sqrt{3}-1, \sqrt{3}-1) \)
(d) \( (\sqrt{3}+1, \sqrt{3}+1) \)
Answer: (b) \( (\sqrt{2}-1, \sqrt{2}-1) \)

Question. The combined equation of straight lines that can be obtained by reflecting the lines \( y = |x - 2| \) in the y-axis is
(a) \( y^2 + x^2 + 4x + 4 = 0 \)
(b) \( y^2 + x^2 - 4x + 4 = 0 \)
(c) \( y^2 - x^2 + 4x - 4 = 0 \)
(d) \( y^2 - x^2 - 4x - 4 = 0 \)
Answer: (d) \( y^2 - x^2 - 4x - 4 = 0 \)

Question. In \( \Delta ABC, B=(0, 0), AB=2, \angle ABC = \frac{\pi}{3} \) and the middle point of BC has the co-ordinates (2, 0). Then the centroid of triangle is
(a) \( \left(\frac{5}{3}, \frac{1}{\sqrt{3}}\right) \)
(b) \( \left(\frac{5}{3}, \frac{1}{\sqrt{3}}\right) \)
(c) \( \left(\frac{5}{\sqrt{3}}, \frac{1}{3}\right) \)
(d) \( \left(\frac{5}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \)
Answer: (b) \( \left(\frac{5}{3}, \frac{1}{\sqrt{3}}\right) \)

Question. In triangle ABC, co-ordinates of A are \( (-1,3) \) and equation of medians and altitude through point B are \( 2x + y = 8 \) and \( 2x + 3y = 8 \) respectively, then
(a) coordinates of C are (4,0)
(b) coordinates of C are (3,9)
(c) coordinates of C are (3,3)
(d) coordinates of centroid are (2,2)
Answer: (b) coordinates of C are (3,9)

Question. The sides of a triangle are \( x + y = 1, 7y = x \) and \( \sqrt{3}y + x = 0 \). Then the following is an interior point of the triangle
(a) Circumcentre
(b) Centroid
(c) Orthocentre
(d) Cannot say
Answer: (b) Centroid

Question. If the equations of the sides of a triangle are \( 2x + y = 2, y = x, \sqrt{3}y + x = 0 \) then which of the following is an exterior point of triangle.
(a) orthocentre
(b) incentre
(c) centroid
(d) Cannot say
Answer: (a) orthocentre

Question. One vertex of the equilateral triangle with centroid at the origin and one side as \( x + y - 2 = 0 \) is
(a) \( (-2, -2) \)
(b) \( (2, 2) \)
(c) \( (2, -2) \)
(d) \( (-2, 2) \)
Answer: (a) \( (-2, -2) \)

Question. A ray of light is sent along the line \( x-2y-3=0 \). On reaching the line \( 3x-2y-5=0 \), the ray is reflected from it. The equation of the line containing the reflected ray.
(a) \( 29x - 2y + 31 = 0 \)
(b) \( 29x + 2y - 31 = 0 \)
(c) \( 29x - 2y - 31 = 0 \)
(d) \( 29x + 2y + 31 = 0 \)
Answer: (c) \( 29x - 2y - 31 = 0 \)

Question. A light ray coming along the line \( 3x + 4y = 5 \) gets reflected from the line \( ax + by = 1 \) and goes along the line \( 5x - 2y = 10 \). Then,
(a) \( a = 64/115, b = 112/15 \)
(b) \( a = 14/15, b = -8/115 \)
(c) \( a = 64/115, b = -8/115 \)
(d) \( a = 64/15, b = 14/15 \)
Answer: (c) \( a = 64/115, b = -8/115 \)

Question. If \( x_1, y_1 \) are roots of \( x^2 + 8x - 20 = 0 \), \( x_2, y_2 \) are the roots of \( 4x^2 + 32x - 57 = 0 \) and \( x_3, y_3 \) are the roots of \( 9x^2 + 72x - 112 = 0 \), then the points \( (x_1, y_1), (x_2, y_2) \) and \( (x_3, y_3) \) where \( x_i < y_i \) for \( i = 1,2,3 \)
(a) are collinear
(b) form an equilateral triangle
(c) form a right angled isosceles triangle
(d) are concyclic
Answer: (a) are collinear

Question. Triangle is formed by the coordinates (0, 0), (0, 21) and (21, 0). The number of integral coordinates strictly inside triangle (integral coordinates has both x and y as integers) :
(a) 190
(b) 105
(c) 231
(d) 205
Answer: (a) 190

Question. Origin is the centre of the square with one of its vertices at (3,4) then the other vertices are
(a) (-3, 4), (-3, -4), (3, -4)
(b) (-4, 3), (-3, -4), (4, -3)
(c) (-4, 3), (-4, -3), (3, -4)
(d) (3, 4), (-4, -3), (4, -3)
Answer: (b) (-4, 3), (-3, -4), (4, -3)

Question. One side of a rectangle lies along the line \( 4x + 7y + 5 = 0 \). Two vertices are (-3,1), (1,1) then the remaining vertices are
(a) \( \left(\frac{1}{65}, \frac{-47}{65}\right), \left(\frac{-131}{65}, \frac{177}{65}\right) \)
(b) \( \left(\frac{-1}{65}, \frac{47}{65}\right), \left(\frac{-131}{65}, \frac{177}{65}\right) \)
(c) \( \left(\frac{1}{65}, \frac{-47}{65}\right), \left(\frac{131}{65}, \frac{-177}{65}\right) \)
(d) (1, -47), (131, 47)
Answer: (a) \( \left(\frac{1}{65}, \frac{-47}{65}\right), \left(\frac{-131}{65}, \frac{177}{65}\right) \)

Question. All points lying inside the triangle formed by the points (1,3), (5,0), (-1,2) satisfy
(a) \( 2x + y - 13 = 0 \)
(b) \( 3x + 2y \geq 0 \)
(c) \( 3x - 4y - 12 \leq 0 \)
(d) \( 4x + y = 0 \)
Answer: (b) \( 3x + 2y \geq 0 \)

Question. If one vertex of an equilateral triangle of side a lies at the origin and the other lies on the line \( x - \sqrt{3}y = 0 \), the coordinates of the third vertex are
(a) \( (0, -a) \)
(b) \( (a, 0) \)
(c) \( \left(\frac{a\sqrt{3}}{2}, \frac{a}{2}\right) \)
(d) \( \left(\frac{a\sqrt{3}}{2}, -\frac{a}{2}\right) \)
Answer: (d) \( \left(\frac{a\sqrt{3}}{2}, -\frac{a}{2}\right) \)

Question. let AB be a line segment of length 4 units with the point A on the line y=2x and B on the line y=x. Then the locus of middle point of all such line segment is
(a) a parabola
(b) an ellipse
(c) a hyperbola
(d) a circle
Answer: (b) an ellipse

Question. If \( (a_1 x + b_1 y + c_1) + (a_2 x + b_2 y + c_2) + (a_3 x + b_3 y + c_3) = 0 \), then the lines \( a_1 x + b_1 y + c_1 = 0 \), \( a_2 x + b_2 y + c_2 = 0 \), \( a_3 x + b_3 y + c_3 = 0 \) can not be parallel
Reason (R): If sum of three straight lines is identically 0 then they are either concurrent or parallel
(a) A and R are true and R is the correct explanation of A
(b) A and R are true and R is not the correct explanation of A
(c) A is true R is False
(d) A is False R is True
Answer: (d) A is False R is True

Question. (A): \( (3,2) \) lies above the line \( x + y + 1 = 0 \)
Reason (R): If the point \( P(x_1, y_1) \) lies above the line \( L = ax + by + c = 0 \), then \( \frac{L(x_1, y_1)}{b} > 0 \)
(a) A and R are true and R is the correct explanation of A
(b) A and R are true and R is not the correct explanation of A
(c) A is true R is False
(d) A is False R is True
Answer: (a) A and R are true and R is the correct explanation of A

Question. Assertion (A): If the angle between the lines \( kx-y+6 = 0 \), \( 3x+5y+7 = 0 \) is \( \pi / 4 \) one value of \( k \) is -4
Reason (R): If \( \theta \) is angle between the lines with slopes \( m_1, m_2 \), then \( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \).
(a) A and R are true and R is the correct explanation of A
(b) A and R are true and R is not the correct explanation of A
(c) A is true R is False
(d) A is False R is True
Answer: (a) A and R are true and R is the correct explanation of A

Question. I : Every first degree equation in \( x \) and \( y \) is \( ax+by+c=0 \), \( |a|+|b| \neq 0 \) represent a straight line
II : Every first degree equation in \( x \) and \( y \) can be convert into slope intercept form
Then which of the following is true
(a) Only I
(b) only II
(c) both I & II
(d) neither I nor II
Answer: (a) Only I

Question. I : Length of the perpendicular from \( (x_1, y_1) \) to the line \( ax+by+c=0 \) is \( \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \)
II : The equation of the line passing through \( (0,0) \) and perpendicular to \( ax+by+c=0 \) is \( bx-ay=0 \)
Then which of the following is true.
(a) only I
(b) only II
(c) both I & II
(d) neither I nor II
Answer: (c) both I & II

Question. I : The ratio in which \( L \equiv ax+by+c=0 \) divides the line segment joining \( A(x_1, y_1) \) \( B(x_2, y_2) \) is \( -\frac{L_{11}}{L_{22}} \)
II: the equation of the line in which \( (x_1, y_1) \) divides the line segment between the coordinate axes in the ratio \( m:n \) is \( \frac{nx}{x_1} + \frac{my}{y_1} = m+n \)
Then which of the following is true
(a) only I
(b) only II
(c) both I & II
(d) neither I nor II
Answer: (c) both I & II

Question. I: A straight line is such that the algebraic sum of the distance from any no. of fixed points is zero. Then that line always passes through a fixed point
II: The base of the triangle lie along the line \( x=a \) and is of length \( a \). If the area of the triangle is \( a^2 \) then the third vertex lies on \( x=-a \) or \( x=3a \).
Then which of the following is true.
(a) only I
(b) only II
(c) both I & II
(d) neither I nor II
Answer: (c) both I & II

Question. Statement I: Normal form of line \( x + y = \sqrt{2} \) is \( x \cos \frac{\pi}{4} + y \sin \frac{\pi}{4} = 1 \)
Statement II: The ratio in which the perpendicular through \( (4,1) \) divides the line joining \( (2,-1) \), \( (6,5) \) is \( 5:8 \)
Which of the above statement (s) is/are true
(a) Only I
(b) Only II
(c) Both I and II
(d) Neither I nor II
Answer: (c) Both I and II

Question. The lines \( L_1 : y - x = 0 \) and \( L_2 : 2x + y = 0 \) intersect the line \( L_3 : y + 2 = 0 \) at P and Q respectively. The bisector of the acute angle between \( L_1 \) and \( L_2 \) intersect \( L_3 \) at R
Statement - 1 : The ratio \( PR : RQ \) equals \( 2\sqrt{2} : \sqrt{5} \).
Statement - 2 : In any triangle, bisector of an angle divides the triangle into two similar triangles. 
(a) Statement – 1 is true, Statement – 2 is true; Statement – 2 is a correct explanation for Statement – 1
(b) Statement – 1 is true, Statement– 2 is false.
(c) Statement – 1 is false, Statement– 2 is true.
(d) Statement – 1 is true, Statement – 2 is true; Statement – 2 is not a correct explanation for Statement – 1
Answer: (b) Statement – 1 is true, Statement– 2 is false.

Question. Observe the following list with respect to the line \( ax+by+c=0 \)
List I
A) Perpendicular distance from \( (0,0) \)
B) X-intercept of the line
C) Y-intercept of the line
D) Circumcentre of triangle OAB where A, B are X and Y intercepts
List II
1) \( -c/b \)
2) \( (-c/a, -c/b) \)
3) \( \frac{|c|}{\sqrt{a^2+b^2}} \)
4) \( \left( -\frac{c}{2a}, -\frac{c}{2b} \right) \)
5) \( -c/a \)

Then the correct answer is
(a) A-3, B-5, C-1, D-2
(b) A-3, B-5, C-1, D-4
(c) A-3, B-4, C-1, D-5
(d) A-1, B-2, C-3, D-4
Answer: (b) A-3, B-5, C-1, D-4

Question. Observe the following
column I
A) the area bounded by the curve \( \text{max}\{|x|, |y|\} = 1 \) is
B) if the point \( (a,a) \) lies between the lines \( |x + y| = 3 \) then number of values of \( [a] \) is (where [.] denotes the greatest integer function)
C) Number of integral values of \( b \) for which the origin and the point \( (1,1) \) lie on the same side of the straight line \( a^2 x + aby + 1 = 0 \) for all \( a \in \mathbb{R} \setminus \{0\} \)
column II
P) 3
Q) 2
R) 4
(a) A → R, B → Q, C → P
(b) A → R, B → P, C → Q
(c) A → P, B → Q, C → R
(d) A → P, B → R, C → Q
Answer: (a) A → R, B → Q, C → P

Question. Equation of line passing through \( (1,3) \), perpendicular to \( 2x-3y+4 = 0 \) is \( ax+by+c = 0 \) (\( a > 0 \)) then ascending order of \( a, b, c \) is
(a) \( a, c, b \)
(b) \( c, b, a \)
(c) \( c, a, b \)
(d) \( a, b, c \)
Answer: (b) \( c, b, a \)

Question. \( A(1,-1), B(4,-1), C(4,3) \) are the vertices of a triangle. Then the equation of the altitude through the vertex ‘\( A \)’ is
(a) \( x = 4 \)
(b) \( y = 4 \)
(c) \( y + 1 = 0 \)
(d) \( x = 1 \)
Answer: (c) \( y + 1 = 0 \)

Question. The equations of the sides of a triangle are \( x-3y=0, 4x+3y=5, 3x+y=0 \). The line \( 3x-4y=0 \) passes through
(a) Incentre
(b) Centroid
(c) Orthocentre
(d) Circumcentre
Answer: (c) Orthocentre

Question. Equation of a diameter of the circum circle of the triangle formed by the lines \( 3x+4y-7=0, 3x-y+5=0 \) and \( 8x-6y+1=0 \) is
(a) \( 3x-y-5=0 \)
(b) \( 3x+y+5=0 \)
(c) \( 3x-y+5=0 \)
(d) \( 3x+y-5=0 \)
Answer: (c) \( 3x-y+5=0 \)

Question. The incentre of the triangle formed by the lines \( x \cos \alpha + y \sin \alpha = \pi, x \cos \beta + y \sin \beta = \pi, x \cos \gamma + y \sin \gamma = \pi \) is \( (\alpha, \beta) \) then \( \alpha + \beta = \)
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (a) 0