Class 11 Mathematics Straight Lines MCQs Set 20

Question. The equations of the lines through \((-1, -1)\) and making angle \(45^\circ\) with the line \(x + y = 0\) are given by
(a) \(x^2 - xy + x - y = 0\)
(b) \(xy - y^2 + x - y = 0\)
(c) \(xy + x + y = 0\)
(d) \(xy + x + y + 1 = 0\)
Answer: (d) \(xy + x + y + 1 = 0\)

Question. The ends of the base of an isosceles triangle are at \((2, 0)\) and \((0, 1)\) and the equation of one side is \(x = 2\) then the orthocentre of the triangle is
(a) \((\frac{3}{2}, \frac{3}{2})\)
(b) \((\frac{5}{4}, 1)\)
(c) \((\frac{3}{4}, 1)\)
(d) \((\frac{4}{3}, \frac{7}{12})\)
Answer: (b) \((\frac{5}{4}, 1)\)

Question. If \(d_1, d_2, d_3\) be perpendiculars from the points \(A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)\) respectively on the line \(x \cos \alpha + y \sin \alpha + \frac{\sin^2 \alpha}{\cos \alpha} = 0\) with \(x_1, x_2, x_3\) are in G.P. and \(y_1, y_2, y_3\) are in A.P. and A, C lie on the curve \(y^2 = 4x\) then \(d_1, d_2, d_3\) are in
(a) A.P
(b) G.P
(c) H.P
(d) A.G.P
Answer: (b) G.P

Question. The equation of the pair of the lines passing through the origin and having slope \(m \in I\) for which equation \((x - 3)(x + m) + 1 = 0\) has integral roots is
(a) \(y^2 - 6xy - 5x^2 = 0\)
(b) \(y^2 - 6xy + 5x^2 = 0\)
(c) \(y^2 + 6xy - 5x^2 = 0\)
(d) \(y^2 + 6xy + 5x^2 = 0\)
Answer: (d) \(y^2 + 6xy + 5x^2 = 0\)

Question. The line \(x + y = 1\) meets the lines represented by the equation \(y^3 - 6xy^2 + 11x^2y - 6x^3 = 0\) at the points P, Q, R. If O is the origin, then \((OP)^2 + (OQ)^2 + (OR)^2\) is equal to
(a) \(\frac{87}{72}\)
(b) \(\frac{121}{72}\)
(c) \(\frac{211}{72}\)
(d) \(\frac{217}{72}\)
Answer: (b) \(\frac{121}{72}\)

Question. Let \(f(x + y) = f(x) \cdot f(y) \,\, \forall x, y \in \mathbb{R}\), \(f(1) = 2\). Area enclosed by \(3|x| + 2|y| \le 8\) is \(\frac{f(k)}{m}\). The point P (2, 6) is translated parallel to \(y = mx\) in the first quadrant through a ‘k’ unit distance. The coordinates of the new position of P is/are
(a) \((2 \pm \frac{2}{\sqrt{10}}, 6 \pm \frac{1}{\sqrt{10}})\)
(b) \((2 \pm \frac{2}{\sqrt{10}}, 6 \pm \frac{18}{\sqrt{10}})\)
(c) \((2 \pm \frac{6}{\sqrt{10}}, 6 \pm \frac{18}{\sqrt{10}})\)
(d) \((2 \pm \frac{6}{\sqrt{10}}, 6 + \frac{18}{\sqrt{10}})\)
Answer: (c) \((2 \pm \frac{6}{\sqrt{10}}, 6 \pm \frac{18}{\sqrt{10}})\)

Question. If \(P = \left( \frac{1}{x_p}, p \right); Q = \left( \frac{1}{x_q}, q \right); R = \left( \frac{1}{x_r}, r \right)\) where \(x_k \neq 0\), denotes the \(k^{th}\) terms of H.P. for \(k \in N\), then
(a) area of \(\Delta PQR = \left( \frac{p^2q^2r^2}{2} \right) \sqrt{(p-q)^2 + (q-r)^2 + (r-p)^2}\)
(b) \(\Delta PQR\) is a right angled triangle.
(c) The points P,Q,R are collinear
(d) None of the options
Answer: (c) The points P,Q,R are collinear

Question. If the lines \(x + y + 1 = 0\); \(4x + 3y + 4 = 0\) and \(x + \alpha y + \beta = 0\), where \(\alpha^2 + \beta^2 = 2\), are concurrent then
(a) \(\alpha = 1, \beta = -1\)
(b) \(\alpha = 1, \beta = \pm 1\)
(c) \(\alpha = -1, \beta = \pm 1\)
(d) \(\alpha = -1, \beta = 1\)
Answer: (d) \(\alpha = -1, \beta = 1\)

Question. Let \(0 < \theta < \frac{\pi}{2}\) be a fixed angle. If \(P(\cos \theta, \sin \theta)\) and \(Q(\cos(\alpha - \theta), \sin(\alpha - \theta))\) then Q is obtained from P by [IIT 2002]
(a) clockwise rotation around origin through an angle \(\alpha\)
(b) anticlockwise rotation around origin through an angle \(\alpha\)
(c) reflection in the line through origin with slope \(\tan \alpha\)
(d) reflection in the line through origin with slope \(\tan \frac{\alpha}{2}\)
Answer: (d) reflection in the line through origin with slope \(\tan \frac{\alpha}{2}\)

Question. The locus of the orthocenter of the triangle formed by the lines \((1 + p)x – py + p(1 + p) = 0\), \((1 + q)x – qy + q(1 + q) = 0\) and \(y = 0\), where \(p \neq q\), is
(a) a hyperbola
(b) a parabola
(c) an ellipse
(d) a straight line
Answer: (d) a straight line

Question. Vertex A of \(\Delta ABC\) moves in such way that \(\tan B + \tan C = a\) constant, when BC is fixed then locus of orthocentre of \(\Delta ABC\) is a
(a) Straight line
(b) Parabola
(c) Ellipse
(d) Circle
Answer: (a) Straight line

Question. Two vertices of a Triangle are \((1, 3)\) and \((4, 7)\). The orthocentre lies on the line \(x + y = 3\). The locus of third vertex is
(a) \(x^2 - 2xy + 2y^2 - 3x - 4y + 36 = 0\)
(b) \(2x^2 - 4xy + 3y^2 - 4x - y + 42 = 0\)
(c) \(3x^2 + xy - 4y^2 - 2x - 24y - 40 = 0\)
(d) \(x^2 - 4xy + 3y^2 - 2x - y - 40 = 0\)
Answer: (c) \(3x^2 + xy - 4y^2 - 2x - 24y - 40 = 0\)

Question. If the line \(ax + by = 1\) passes through point of intersection of \(y = x \tan \alpha + p \sec \alpha\); \(y \sin(30^\circ - \alpha) - x \cos(30^\circ - \alpha) = p\) and is inclined at \(30^\circ\) with \(y = \tan \alpha\), then \(a^2 + b^2 =\)
(a) \(\frac{1}{p^2}\)
(b) \(\frac{2}{p^2}\)
(c) \(\frac{3}{2p^2}\)
(d) \(\frac{3}{4p^2}\)
Answer: (d) \(\frac{3}{4p^2}\)

Question. Let \(A(6, 7), B(2, 3), C(-2, 1)\) vertices of a Triangle. The point P in the interior of \(\Delta ABC\) such that \(\Delta PBC\) is an equilateral triangle is
(a) \((-\sqrt{3}, 2 + 2\sqrt{3})\)
(b) \((-\sqrt{3}, 2 - 2\sqrt{3})\)
(c) \((\sqrt{3}, 2 - 2\sqrt{3})\)
(d) \((\sqrt{3}, 2 + 2\sqrt{3})\)
Answer: (a) \((-\sqrt{3}, 2 + 2\sqrt{3})\)

Question. In an Equilateral triangle ex-centre opposite vertex A is \((2, -4)\) and equation of side BC is \(x + y - 2 = 0\). If coordinates of vertex A is \((\alpha, \beta)\) then \(2\alpha - \beta\)
(a) 12
(b) 6
(c) -6
(d) 4
Answer: (a) 12

Question. A variable line whose slope is \(-2\) cuts x and y axes respectively at points A and C. A Rhombus \(ABCD\) is completed, such that the vertex B lies on the line \(y = x\), then the locus of the vertex D is
(a) \(x + y + 2 = 0\)
(b) \(x + y + 3 = 0\)
(c) \(x + y = 0\)
(d) \(x + y - 1 = 0\)
Answer: (c) \(x + y = 0\)

Question. The length of line segment joining the feet of the perpendiculars drawn from the point \((3, 4)\) on the pair of lines \(x^2 - 5xy + 6y^2 = 0\) is
(a) \(\sqrt{2}\)
(b) \(2\)
(c) \(\frac{1}{\sqrt{2}}\)
(d) \(2\sqrt{2}\)
Answer: (c) \(\frac{1}{\sqrt{2}}\)

Question. The number of possible straight lines, passing through \((2, 3)\) and forming a triangle with coordinates axes, whose area is \(12\) sq. units, is:
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. Given \(A(0, 0)\) and \(B(x, y)\) with \(x \in (0, 1)\) and \(y > 0\). Let the slope of the line AB equals \(m_1\). Point C lies on the line \(x = 1\) such that the slope of BC equals \(m_2\) where \(0 < m_2 < m_1\). If the area of the triangle ABC can be expressed as \((m_1 - m_2) f(x)\), then the largest possible value of \(f(x)\) is :
(a) 1
(b) 1/2
(c) 1/4
(d) 1/8
Answer: (d) 1/8

MULTI ANSWER QUESTIONS

Question. If one diagonal of a square is the portion of line \(\frac{x}{a} + \frac{y}{b} = 1\) intercepted by the axes, then the extremities of the other diagonal of the square are
(a) \((\frac{a+b}{2}, \frac{a+b}{2})\)
(b) \((\frac{a-b}{2}, \frac{a+b}{2})\)
(c) \((\frac{a-b}{2}, \frac{b-a}{2})\)
(d) \((\frac{a+b}{2}, \frac{b-a}{2})\)
Answer: (a) \((\frac{a+b}{2}, \frac{a+b}{2})\) and (c) \((\frac{a-b}{2}, \frac{b-a}{2})\)

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

Question. Let \(B(1, -3)\) and \(D(0, 4)\) represents two vertices of a rhombus \(ABCD\) in xy plane, then coordinates of vertex A if \(\angle BAD = 60^\circ\) can be equal to:
(a) \((\frac{1-7\sqrt{3}}{2}, \frac{1-\sqrt{3}}{2})\)
(b) \((\frac{1+7\sqrt{3}}{2}, \frac{1+\sqrt{3}}{2})\)
(c) \((\frac{-1+7\sqrt{3}}{2}, \frac{-1+\sqrt{3}}{2})\)
(d) \((\frac{-1-7\sqrt{3}}{2}, \frac{-1-\sqrt{3}}{2})\)
Answer: (a) \((\frac{1-7\sqrt{3}}{2}, \frac{1-\sqrt{3}}{2})\) and (b) \((\frac{1+7\sqrt{3}}{2}, \frac{1+\sqrt{3}}{2})\)

Question. If \(6a^2 + 12b^2 + 2c^2 + 17ab - 10bc - 7ac = 0\) then all the lines represented by \(ax + by + c = 0\) are concurrent at the point
(a) \((-2, -3)\)
(b) \((2, 3)\)
(c) \((-\frac{3}{2}, -2)\)
(d) \((-\frac{3}{2}, 2)\)
Answer: (a) \((-2, -3)\) and (c) \((-\frac{3}{2}, -2)\)

Question. The point \(A(0, 0)\), \(B(\cos \alpha, \sin \alpha)\) and \(C(\cos \beta, \sin \beta)\) are the vertices of a right angled triangle if ;
(a) \(\sin(\frac{\alpha - \beta}{2}) = \frac{1}{\sqrt{2}}\)
(b) \(\cos(\frac{\alpha - \beta}{2}) = -\frac{1}{\sqrt{2}}\)
(c) \(\cos(\frac{\alpha - \beta}{2}) = \frac{1}{\sqrt{2}}\)
(d) \(\sin(\frac{\alpha - \beta}{2}) = -\frac{1}{\sqrt{2}}\)
Answer: (a), (b), (c), (d) are all possible

Question. Let \(A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)\) be the vertices of the triangle ABC such that \(x_3 = lx_1 + mx_2, y_3 = ly_1 + my_2\) where \(l, m < 0\) then, origin
(a) Lies out side triangle ABC
(b) And C lies on the same side of AB
(c) May lie inside triangle ABC
(d) None of the options
Answer: (b) And C lies on the same side of AB and (c) May lie inside triangle ABC

Question. Two sides of a rhombus \(OABC\) (lying entirely in \(Q_1\) or \(Q_3\)) of area equal to 2 square units are \(y = \frac{x}{\sqrt{3}}, y = \sqrt{3}x\). The possible coordinates of B is / are (‘O’ is origin)
(a) \((1 + \sqrt{3}, 1 + \sqrt{3})\)
(b) \((-1 - \sqrt{3}, -1 - \sqrt{3})\)
(c) \((\sqrt{3} - 1, \sqrt{3} - 1)\)
(d) \((1 - \sqrt{3}, 1 - \sqrt{3})\)
Answer: (a) \((1 + \sqrt{3}, 1 + \sqrt{3})\) and (b) \((-1 - \sqrt{3}, -1 - \sqrt{3})\)

Question. (A) : If \( (a_1x + b_1y + c_1) + (a_2x + b_2y + c_2) + (a_3x + b_3y + c_3) = 0 \), then the lines \( a_1x + b_1y + c_1 = 0, a_2x + b_2y + c_2 = 0, a_3x + b_3y + c_3 = 0 \) can not be parallel
(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: \( (3,2) \) is lies above the line \( x + y + 1 = 0 \)
R: If the point \( P(x_1, y_1) \) lies above the line \( L = ax + by + c \) 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 \( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| \).
(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 \( A(x_1, y_1) \) and \( B(x_2, y_2) \) 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) devides 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

COMPREHENSION QUESTIONS

PASSAGE : I

A (1, 3) and \( C \left( -\frac{2}{5}, -\frac{2}{5} \right) \) are the vertices of a triangle ABC and the equation of the angle bisector of \( \angle ABC \) is \( x + y = 2 \)

Question. Equation of BC is
(a) \( 7x + 3y + 4 = 0 \)
(b) \( 3x + 7y + 4 = 0 \)
(c) \( 13x + 7y + 8 = 0 \)
(d) \( x + 9y + 4 = 0 \)
Answer: (a) \( 7x + 3y + 4 = 0 \)

Question. Coordinates of vertex B
(a) \( \left( \frac{3}{10}, \frac{17}{10} \right) \)
(b) \( \left( \frac{17}{10}, \frac{3}{10} \right) \)
(c) \( \left( -\frac{5}{2}, \frac{9}{2} \right) \)
(d) (1, 1)
Answer: (c) \( \left( -\frac{5}{2}, \frac{9}{2} \right) \)

Question. Equation of side AB is
(a) \( 13x - 7y + 8 = 0 \)
(b) \( 13x + 7y - 34 = 0 \)
(c) \( 3x + 7y - 24 = 0 \)
(d) \( 3x + 7y + 24 = 0 \)
Answer: (c) \( 3x + 7y - 24 = 0 \)

PASSAGE : II

The base of an isosceles triangle is equal to 4 units, the base angle is equal to \( 45^\circ \). A straight line cuts the extension of the base at a point M at the angle \( \theta \) and bisects the lateral side of the triangle which is nearest to M.

Question. The area of quadrilateral which the straight line cuts off from the given triangle is
(a) \( \frac{3 + \tan \theta}{1 + \tan \theta} \)
(b) \( \frac{3 + 2\tan \theta}{1 + \tan \theta} \)
(c) \( \frac{3 + \tan \theta}{1 - \tan \theta} \)
(d) \( \frac{3 + 5\tan \theta}{1 + \tan \theta} \)
Answer: (d) \( \frac{3 + 5\tan \theta}{1 + \tan \theta} \)

Question. The possible range of values in which area of quadrilateral which straight line cuts off from the given triangle lie in
(a) \( \left( \frac{5}{2}, \frac{7}{2} \right) \)
(b) (4, 5)
(c) \( \left( 4, \frac{9}{2} \right) \)
(d) (3, 4)
Answer: (d) (3, 4)

Question. The length of portion of straight line inside the triangle may lie in the range
(a) \( (2, 4) \)
(b) \( \left( \frac{3}{2}, \sqrt{3} \right) \)
(c) \( (\sqrt{2}, 2) \)
(d) \( (\sqrt{2}, \sqrt{3}) \)
Answer: (c) \( (\sqrt{2}, 2) \)