Class 11 Mathematics Straight Lines MCQs Set 23

Question. The lines \( p(p^2 + 1)x - y + q = 0 \) and \( (p^2 + 1)^2 x + (p^2 + 1)y + 2q = 0 \) are perpendicular to a common line for
(a) exactly one value of p
(b) exactly two values of p
(c) more than two values of p
(d) no values of p
Answer: (a) exactly one value of p

Question. The slope of the line passing through the points \( (2, \sin \theta) \), \( (1, \cos \theta) \) is 0 then general solution of \( \theta \)
(a) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)
(b) \( n\pi - \frac{\pi}{4}, \forall n \in Z \)
(c) \( n\pi \pm \frac{\pi}{4}, \forall n \in Z \)
(d) \( n\pi, \forall n \in Z \)
Answer: (a) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)

SLOPE-INTERCEPT FORM, SLOPE-POINT FORM AND TWO-POINT FORM

Question. The perpendicular bisector of the line segment joining \( P(1, 4) \) and \( Q(K, 3) \) has Y intercept -4. then a possible value of K is
(AIEEE-2008)
(a) -4
(b) 1
(c) 2
(d) -2
Answer: (a) -4

Question. \( P(\alpha, \beta) \) lies on the line \( y = 6x - 1 \) and \( Q(\beta, \alpha) \) lies on the line \( 2x - 5y = 5 \). Then the equation of the line \( PQ \) is
(a) \( 2x + y = 3 \)
(b) \( 3x + 2y = 5 \)
(c) \( x + y = 6 \)
(d) \( 3x + y = 7 \)
Answer: (c) \( x + y = 6 \)

Question. A line joining \( A(2,0) \) and \( B(3,1) \) is rotated about A in anticlock wise direction through angle \( 15^\circ \), then the equation of AB in the new position is
(a) \( y = \sqrt{3}x - 2 \)
(b) \( y = \sqrt{3}(x - 2) \)
(c) \( y = \sqrt{3}(x + 2) \)
(d) \( x - 2 = \sqrt{3}y \)
Answer: (b) \( y = \sqrt{3}(x - 2) \)

INTERCEPTS AND INTERCEPT FORM

Question. The line \( 2x + 3y = 6, 2x + 3y = 8 \) cut the X-axis at A, B respectively. A line \( L = 0 \) drawn through the point (2,2) meets the X-axis at C in such a way that abscissa of A, B, C are in arithmetic Progression. then the equation of the line L is
(a) \( 2x + 3y = 10 \)
(b) \( 3x + 2y = 10 \)
(c) \( 2x - 3y = 10 \)
(d) \( 3x - 2y = 10 \)
Answer: (a) \( 2x + 3y = 10 \)

Question. The sum of the intercepts cut off by the axes on lines \( x + y = a \), \( x + y = ar \), \( x + y = ar^2 \), ............ where \( a \neq 0 \) and \( r = \frac{1}{2} \)
(a) \( 2a \)
(b) \( a\sqrt{2} \)
(c) \( 2\sqrt{2}a \)
(d) \( a \)
Answer: (c) \( 2\sqrt{2}a \)

Question. The equation of the straight line which bisects the intercepts between the axes of the lines \( x + y = 2 \) and \( 2x + 3y = 6 \) is
(a) \( 2x = 3 \)
(b) \( y = 1 \)
(c) \( 2y = 3 \)
(d) \( x = 1 \)
Answer: (b) \( y = 1 \)

Question. Equation of the line passing through (0, 1) and having intercepts in the ratio \( 2 : 3 \) is
(a) \( 2x + 3y = 3 \)
(b) \( 2x - 3y + 3 = 0 \)
(c) \( 3x + 2y = 2 \)
(d) \( 2x - 3y - 3 = 0 \)
Answer: (c) \( 3x + 2y = 2 \)

NORMAL FORM AND SYMMETRIC FORM

Question. A straight line is such that its distance of 5 units from the origin and its inclination is \( 135^\circ \). The intercepts of the line on the co-ordinate axes are
(a) 5, 5
(b) \( \sqrt{2}, \sqrt{2} \)
(c) \( 5\sqrt{2}, 5\sqrt{2} \)
(d) \( 5/\sqrt{2}, 5/\sqrt{2} \)
Answer: (c) \( 5\sqrt{2}, 5\sqrt{2} \)

Question. Angles made with the x - axis by two lines drawn through the point (1, 2) and cutting the line \( x + y = 4 \) at a distance \( \sqrt{\frac{2}{3}} \) from the point (1,2) are
(a) \( \frac{\pi}{6} \) and \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{8} \) and \( \frac{3\pi}{8} \)
(c) \( \frac{\pi}{12} \) and \( \frac{5\pi}{12} \)
(d) \( \frac{\pi}{4} \) and \( \frac{\pi}{2} \)
Answer: (c) \( \frac{\pi}{12} \) and \( \frac{5\pi}{12} \)

PROBLEMS ON DISTANCES

Question. Perpendicular distance from the origin to the line joining the points \( (a\cos \theta, a\sin \theta) \) and \( (a\cos \phi, a\sin \phi) \) is
(a) \( 2a \cos(\theta - \phi) \)
(b) \( a \cos\left(\frac{\theta - \phi}{2}\right) \)
(c) \( 4a \cos\left(\frac{\theta - \phi}{2}\right) \)
(d) \( a \cos\left(\frac{\theta + \phi}{2}\right) \)
Answer: (b) \( a \cos\left(\frac{\theta - \phi}{2}\right) \)

Question. One side of an equilateral triangle is \( 3x + 4y = 7 \) and its vertex is (1,2). Then the length of the side of the triangle is
(a) \( \frac{4\sqrt{3}}{17} \)
(b) \( \frac{3\sqrt{3}}{16} \)
(c) \( \frac{8\sqrt{3}}{15} \)
(d) \( \frac{4\sqrt{3}}{15} \)
Answer: (c) \( \frac{8\sqrt{3}}{15} \)

Question. Equation of the line through the point of intersection of the lines \( 3x + 2y + 4 = 0 \) and \( 2x + 5y - 1 = 0 \) whose distance from (2,-1) is 2.
(a) \( 2x - y + 5 = 0 \)
(b) \( 4x + 3y + 5 = 0 \)
(c) \( x + 2 = 0 \)
(d) \( 3x + y + 5 = 0 \)
Answer: (b) \( 4x + 3y + 5 = 0 \)

Question. If p, q denote the lengths of the perpendiculars from the origin on the lines \( x \sec \alpha - y \csc \alpha = a \) and \( x \cos \alpha + y \sin \alpha = a \cos 2\alpha \) then
am (a) \( 4p^2 + q^2 = a^2 \)
(b) \( p^2 + q^2 = a^2 \)
(c) \( p^2 + 2q^2 = a^2 \)
(d) \( 4p^2 + q^2 = 2a^2 \)
Answer: (a) \( 4p^2 + q^2 = a^2 \)

Question. The distance between two parallel lines is \( p_1 - p \). The equation of one line is \( x \cos \alpha + y \sin \alpha = p \) then the equation of the 2nd line is
(a) \( x \cos \alpha + y \sin \alpha + p_1 + 2p = 0 \)
(b) \( x \cos \alpha + y \sin \alpha = 2p_1 - p \)
(c) \( x \cos \alpha + y \sin \alpha = 0 \)
(d) \( x \cos \alpha + y \sin \alpha + p_1 - 2p = 0 \)
Answer: (d) \( x \cos \alpha + y \sin \alpha + p_1 - 2p = 0 \)

Question. The ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \) is
(a) 7 : 3
(b) 3 : 7
(c) 2 : 3
(d) 3 : 4
Answer: (b) 3 : 7

Question. The equations of the lines parallel to \( 4x + 3y + 2 = 0 \) and at a distance of ‘4’ units from it are
(a) \( 4x + 3y + 22 = 0, 4x + 3y - 20 = 0 \)
(b) \( 4x + 3y + 22 = 0, 4x + 3y - 18 = 0 \)
(c) \( 4x + 3y - 18 = 0, 4x + 3y - 20 = 0 \)
(d) \( 4x - 3y - 18 = 0, 4x + 3y - 20 = 0 \)
Answer: (b) \( 4x + 3y + 22 = 0, 4x + 3y - 18 = 0 \)

POSITION OF A POINT (S) W.R.T. LINE (S)

Question. The range of \( \alpha \) for which the points \( (\alpha, \alpha + 2) \) and \( \left(\frac{3\alpha}{2}, \alpha^2\right) \) lie on opposite sides of the line \( 2x + 3y - 6 = 0 \)
(a) \( (-\infty, -2) \)
(b) \( (0, 1) \)
(c) \( (-\infty, -2) \cup (0, 1) \)
(d) \( (-\infty, 1) \cup (2, \infty) \)
Answer: (c) \( (-\infty, -2) \cup (0, 1) \)

Question. If \( P\left(1 + \frac{t}{\sqrt{2}}, 2 + \frac{t}{\sqrt{2}}\right) \) be any point on a line then the range of values of t for which the point P lies between the parallel lines \( x + 2y = 1 \) and \( 2x + 4y = 15 \) is
(a) \( -\frac{4\sqrt{2}}{5} < t < \frac{5\sqrt{2}}{6} \)
(b) \( -\frac{4\sqrt{2}}{3} < t < \frac{5\sqrt{2}}{6} \)
(c) \( t < -\frac{4\sqrt{2}}{3} \)
(d) \( t < \frac{5\sqrt{2}}{6} \)
Answer: (b) \( -\frac{4\sqrt{2}}{3} < t < \frac{5\sqrt{2}}{6} \)

Question. A point which lies between \( 2x + 3y - 7 = 0 \) and \( 2x + 3y + 12 = 0 \) is
(a) (5, 1)
(b) (-1, 3)
(c) (3, -5)
(d) (7, -1)
Answer: (c) (3, -5)

Question. A line L cuts the sides AB, BC of \( \Delta ABC \) in the ratio \( 2 : 5 \), \( 7 : 4 \) respectively. Then the line L cuts CA in the ratio
(a) 7 : 10
(b) 7 : –10
(c) 10 : 7
(d) 10 : –7
Answer: (d) 10 : –7

POINT OF INTERSECTION OF LINES AND CONCURRENCY OF LINES

Question. The number of integral values of m for which x-coordinate of point of intersection of the lines \( 3x + 4y = 9 \) and \( y = mx + 1 \) is also an integer is
(a) 2
(b) 0
(c) 4
(d) 11
Answer: (a) 2

Question. The line parallel to the x-axis and passing through the intersection of the lines \( ax + 2by + 3b = 0 \) and \( bx - 2ay - 3a = 0 \), where \( (a, b) \neq (0, 0) \) is
(a) Above the x-axis at a distance of 3/2 from it
(b) Above the x-axis at a distance of 2/3 from it
(c) Below the x-axis at a distance of 3/2 from it
(d) Below the x-axis at a distance of 2/3 from it
Answer: (c) Below the x-axis at a distance of 3/2 from it

Question. If a, b, c form a G P with common ratio r, the sum of the ordinates of the points of intersection of the line \( ax + by + c = 0 \) and the curve \( x + 2y^2 = 0 \) is
(a) \( \frac{-r}{2} \)
(b) \( \frac{-r^2}{2} \)
(c) \( \frac{r}{2} \)
(d) \( \frac{r^2}{2} \)
Answer: (c) \( \frac{r}{2} \)

Question. Consider a family of straight lines \( (x + y) + \lambda(2x - y + 1) = 0 \). Find the equation of the straight line belonging to this family that is farthest from (1, –3).
(a) \( 3x - 3y + 2 = 0 \)
(b) \( 6x + 15y - 7 = 0 \)
(c) \( 5x + 2y + 1 = 0 \)
(d) \( 6x - 15y + 7 = 0 \)
Answer: (d) \( 6x - 15y + 7 = 0 \)

Question. If the line \( x = a + m \), \( y = -2 \) and \( y = mx \) are concurrent, then least value of \( |a| \) is
(a) 0
(b) \( \sqrt{2} \)
(c) \( 2\sqrt{2} \)
(d) 2
Answer: (c) \( 2\sqrt{2} \)

Question. If \( a \neq b \neq c \), if \( ax + by + c = 0 \), \( bx + cy + a = 0 \) and \( cx + ay + b = 0 \) are concurrent. Then the value of \( 2^{a^2 b^{-1} c^{-1}} \cdot 2^{b^2 c^{-1} a^{-1}} \cdot 2^{c^2 a^{-1} b^{-1}} \)
(a) 1
(b) 4
(c) 8
(d) 16
Answer: (c) 8

Question. Line \( ax + by + p = 0 \) makes angle \( \pi / 4 \) with \( x \cos \alpha + y \sin \alpha = p, p \in R^+ \). If these lines and the line \( x \sin \alpha - y \cos \alpha = 0 \) are concurrent, then
(a) \( a^2 + b^2 = 1 \)
(b) \( a^2 + b^2 = 2 \)
(c) \( 2(a^2 + b^2) = 1 \)
(d) \( a^2 - b^2 = 1 \)
Answer: (b) \( a^2 + b^2 = 2 \)

ANGLE BETWEEN LINES

Question. If p, q, r are distinct, then \( (q - r)x + (r - p)y + (p - q) = 0 \) and \( (q^3 - r^3)x + (r^3 - p^3)y + (p^3 - q^3) = 0 \) represents the same line if
(a) \( p + q + r = 0 \)
(b) \( p = q = r \)
(c) \( p^2 + q^2 + r^2 = 0 \)
(d) \( p^3 + q^3 + r^3 = 0 \)
Answer: (a) \( p + q + r = 0 \)

Question. The lines \( (a + b)x + (a - b)y - 2ab = 0 \), \( (a - b)x + (a + b)y - 2ab = 0 \) and \( x + y = 0 \) form an isosceles triangle whose vertical angle is
(a) \( \frac{\pi}{2} \)
(b) \( \tan^{-1}\left(\frac{2ab}{a^2 - b^2}\right) \)
(c) \( \tan^{-1}\left(\frac{a}{b}\right) \)
(d) \( 2\tan^{-1}\left(\frac{a}{b}\right) \)
Answer: (b) \( \tan^{-1}\left(\frac{2ab}{a^2 - b^2}\right) \)

Question. If \( 2(\sin a + \sin b)x - 2\sin(a - b)y = 3 \) and \( 2(\cos a + \cos b)x + 2\cos(a - b)y = 5 \) are perpendicular then \( \sin 2a + \sin 2b = \)
(a) \( \sin(a - b) - 2\sin(a + b) \)
(b) \( \sin 2(a - b) - 2\sin(a + b) \)
(c) \( 2\sin(a - b) - \sin(a + b) \)
(d) \( \sin 2(a - b) - \sin(a + b) \)
Answer: (b) \( \sin 2(a - b) - 2\sin(a + b) \)

Question. Two equal sides of an isosceles triangle are given by \( 7x - y + 3 = 0 \) and \( x + y - 3 = 0 \) and the third side passes through the point (1, 10) then the slope m of the third side is given by
(a) \( 3m^2 - 1 = 0 \)
(b) \( m^2 + 1 = 0 \)
(c) \( 3m^2 + 8m - 3 = 0 \)
(d) \( m^2 - 3 = 0 \)
Answer: (c) \( 3m^2 + 8m - 3 = 0 \)

Question. The diagonal of a square is \( 8x - 15y = 0 \) and one vertex of the square is (1, 2). Then the equations to the sides of the square passing through the vertex are
(a) \( 22x + 8y = 9, 22x - 8y = 52 \)
(b) \( 23x + 7y = 9, 7x - 23y = 52 \)
(c) \( 23x - 7y = 9, 7x + 23y = 53 \)
(d) \( 22x - 8y = 9, 22x + 8y = 52 \)
Answer: (c) \( 23x - 7y = 9, 7x + 23y = 53 \)

TRIANGLES AND AREA OF THE TRIANGLE

Question. Area of triangle formed by angle bisectors of coordinate axes and the line \( x = 6 \) in sq.units is
(a) 36
(b) 18
(c) 72
(d) 9
Answer: (a) 36

Question. The quadratic equation whose roots are the x and y intercepts of the line passing through (1,1) and making a triangle of area A with the co -ordinate axes is
(a) \( x^2 + Ax + 2A = 0 \)
(b) \( x^2 - 2Ax + 2A = 0 \)
(c) \( x^2 - Ax + 2A = 0 \)
(d) \( (x - A)(x + A) = 0 \)
Answer: (b) \( x^2 - 2Ax + 2A = 0 \)

Question. A line passing through (3,4) meets the axes OX and OY at A and B respectively. The minimum area of the triangle OAB in square units is
(a) 8
(b) 16
(c) 24
(d) 32
Answer: (c) 24

QUADRILATERALS AND AREA OF THE QUADRILATERALS

Question. The figure formed by the straight lines \( \sqrt{3}x + y = 0 \), \( \sqrt{3}y + x = 0 \), \( \sqrt{3}x + y = 1 \), \( \sqrt{3}y + x = 1 \) is
(a) a rectangle
(b) a square
(c) a rhombus
(d) parallelogram
Answer: (c) a rhombus

Question. Let the base of a triangle lie along the line \( x = a \) and be of length a. The area of this triangle is \( a^2 \), if the vertex lies on the line
(a) \( x + a = 0 \)
(b) \( x = 0 \)
(c) \( 2x - a = 0 \)
(d) \( x - a = 0 \)
Answer: (a) \( x + a = 0 \)

Question. The area bounded by \( y = |x| - 1 \), \( y = -|x| + 1 \)
(a) 1
(b) 2
(c) \( 2\sqrt{2} \)
(d) 4
Answer: (b) 2