JEE Mathematics Coordinates and Straight Lines MCQs Set 02

Choose the most appropriate option (a, b, c or d)

Question. If p and p' are the perpendiculars from the origin upon the lines \( x \sec \theta + y \csc \theta = a \) and \( x \cos \theta - y \sin \theta = a \cos 2\theta \) respectively then
(a) \( 4p^2 + p'^2 = a^2 \)
(b) \( p^2 + 4p'^2 = a^2 \)
(c) \( p^2 + p'^2 = a^2 \)
(d) none of the options
Answer: (a) \( 4p^2 + p'^2 = a^2 \)

Question. Let the perpendiculars from any point on the line \( 2x + 11y = 5 \) upon the lines \( 24x + 7y = 20 \) and \( 4x – 3y = 2 \) have the lengths p and p' respectively. Then
(a) \( 2p = p' \)
(b) \( p = p' \)
(c) \( p = 2p' \)
(d) none of the options
Answer: (b) \( p = p' \)

Question. If \( P(1 + t/\sqrt{2}) \) 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) \( 0 < t < \frac{5\sqrt{2}}{6} \)
(c) \( -\frac{4\sqrt{2}}{5} < t < 0 \)
(d) none of the options
Answer: (a) \( -\frac{4\sqrt{2}}{5} < t < \frac{5\sqrt{2}}{6} \)

Question. There are two parallel lines, one of which has the equation \( 3x + 4y = 2 \). If the lines cut an intercept of length 5 on the line \( x + y = 1 \) then the equation of the other line is
(a) \( 3x + 4y = \frac{\sqrt{6} - 2}{2} \)
(b) \( 3x + 4y = \frac{\sqrt{6} + 2}{2} \)
(c) \( 3x + 4y = 7 \)
(d) none of the options
Answer: (d) none of the options

Question. If the intercept made on the line \( y = mx \) by lines \( y = 2 \) and \( y = 6 \) is less than 5 then the range of values of m is
(a) \( \left( -\infty, -\frac{4}{3} \right) \cup \left( \frac{4}{3}, +\infty \right) \)
(b) \( \left( -\frac{4}{3}, \frac{4}{3} \right) \)
(c) \( \left( -\frac{3}{4}, \frac{3}{4} \right) \)
(d) none of the options
Answer: (a) \( \left( -\infty, -\frac{4}{3} \right) \cup \left( \frac{4}{3}, +\infty \right) \)

Question. If \( a, b, c \) are any three terms of an AP then the line \( ax + by + c = 0 \)
(a) has a fixed direction
(b) always passes through a fixed point
(c) always cuts intercepts on the axes such that their sum is zero
(d) forms a triangle with the axes whose area is constant
Answer: (b) always passes through a fixed point

Question. If \( a, c, b \) are in GP then the line \( ax + by + c = 0 \)
(a) has a fixed direction
(b) always passes through a fixed point
(c) forms a triangle with the axes whose area is constant
(d) always cuts intercepts on the axes such that their sum is zero
Answer: (c) forms a triangle with the axes whose area is constant

Question. The number of real values of k for which the lines \( x – 2y + 3 = 0 \), \( kx + 3y + 1 = 0 \) and \( 4x – ky + 2 = 0 \) are concurrent is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (a) 0

Question. A family of lines is given by \( (1 + 2\lambda)x + (1 - \lambda)y + \lambda = 0 \), \( \lambda \) being the parameter. The line belonging to this family at the maximum distance from the point \( (1, 4) \) is
(a) \( 4x – y + 1 = 0 \)
(b) \( 33x + 12y + 7 = 0 \)
(c) \( 12x + 33y = 7 \)
(d) none of the options
Answer: (c) \( 12x + 33y = 7 \)

Question. The members of the family of lines \( (\lambda + \mu)x + (2\lambda + \mu)y = \lambda + 2\mu \), where \( \lambda \neq 0 \), \( \mu \neq 0 \), pass through the point
(a) \( (3, -1) \)
(b) \( (-3, 1) \)
(c) \( (1, 1) \)
(d) none of the options
Answer: (a) \( (3, -1) \)

Question. The equations of the sides AB, BC and CA of the \( \Delta ABC \) are \( y – x = 2 \), \( x + 2y = 1 \) and \( 3x + y + 5 = 0 \) respectively. The equation of the altitude through B is
(a) \( x – 3y + 1 = 0 \)
(b) \( x – 3y + 4 = 0 \)
(c) \( 3x – y + 2 = 0 \)
(d) none of the options
Answer: (b) \( x – 3y + 4 = 0 \)

Question. The range of values of the ordinate of a point moving on the line \( x = 1 \), and always remaining in the interior of the triangle formed by the lines \( y = x \), the x-axis and \( x + y = 4 \), is
(a) \( (0, 1) \)
(b) \( [0, 1] \)
(c) \( [0, 4] \)
(d) none of the options
Answer: (a) \( (0, 1) \)

Question. If the point \( (a, a) \) falls between the lines \( |x + y| = 2 \) then
(a) \( |a| = 2 \)
(b) \( |a| = 1 \)
(c) \( |a| < 1 \)
(d) \( |a| < \frac{1}{2} \)
Answer: (c) \( |a| < 1 \)

Question. If \( A(\sin \alpha, 1/\sqrt{2}) \) and \( B(1/\sqrt{2}, \cos \alpha) \), \( -\pi \le \alpha \le \pi \), are two point on the same side of the line \( x – y = 0 \) then \( \alpha \) belongs to the interval
(a) \( \left( -\frac{\pi}{4}, \frac{\pi}{4} \right) \cup \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \)
(b) \( \left( -\frac{\pi}{4}, \frac{\pi}{4} \right) \)
(c) \( \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \)
(d) none of the options
Answer: (a) \( \left( -\frac{\pi}{4}, \frac{\pi}{4} \right) \cup \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \)

Question. The straight lines \( L_1 \equiv 4x – 3y + 2 = 0 \), \( L_2 \equiv 3x + 4y – 4 = 0 \) and \( L_3 \equiv x – 7y + 6 = 0 \)
(a) form a right-angled triangle
(b) from a right-angled isosceles triangle
(c) are concurrent
(d) none of the options
Answer: (c) are concurrent

Question. The equation of bisector of the acute angle between the lines \( 2x – y + 4 = 0 \) and \( x – 2y = 1 \) is
(a) \( x + y + 5 = 0 \)
(b) \( x – y + 1 = 0 \)
(c) \( x – y = 5 \)
(d) none of the options
Answer: (b) \( x – y + 1 = 0 \)

Question. The equation of the bisector of the acute angle between the lines \( 2x – y + 4 = 0 \) and \( x – 2y = 1 \) is
(a) \( (\sqrt{5} - 2\sqrt{2})x + (\sqrt{5} + \sqrt{2})y = 3\sqrt{5} - 2\sqrt{2} \)
(b) \( (\sqrt{5} + 2\sqrt{2})x + (\sqrt{5} - \sqrt{2})y = 3\sqrt{5} + 2\sqrt{2} \)
(c) \( 3x = 10 \)
(d) none of the options
Answer: (a) \( (\sqrt{5} - 2\sqrt{2})x + (\sqrt{5} + \sqrt{2})y = 3\sqrt{5} - 2\sqrt{2} \)

Question. Two lines \( 2x – 3y = 1 \) and \( x + 2y + 3 = 0 \) divide the x-y plane in four compartments which are named as shown in figure. Consider the locations of the points \( (2, -1) \), \( (3, 2) \) and \( (-1, -2) \). We get
(a) \( (2, -1) \in IV \)
(b) \( (3, 2) \in III \)
(c) \( (-1, -2) \in II \)
(d) none of the options
Answer: (a) \( (2, -1) \in IV \)

Question. If the lines \( y – x = 5 \), \( 3x + 4y = 1 \) and \( y = mx + 3 \) are concurrent then the value of m is
(a) \( \frac{19}{5} \)
(b) 1
(c) \( \frac{5}{19} \)
(d) none of the options
Answer: (c) \( \frac{5}{19} \)

Question. If the point \( (\cos \theta, \sin \theta) \) does not fall in that angle between the lines \( y = |x – 1| \) in which the origin lies then \( \theta \) belongs to
(a) \( \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \)
(b) \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)
(c) \( (0, \pi) \)
(d) none of the options
Answer: (b) \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)

Question. The points \( (-1, 1) \) and \( (1, -1) \) are symmetrical about the line
(a) \( y + x = 0 \)
(b) \( y = x \)
(c) \( x + y = 1 \)
(d) none of the options
Answer: (b) \( y = x \)

Question. The equations of the line segment AB is \( y = x \). If A and B lie on the same side of the line mirror \( 2x – y = 1 \), the image of AB has the equation
(a) \( x + y = 2 \)
(b) \( 8x + y = 9 \)
(c) \( 7x – y = 6 \)
(d) none of the options
Answer: (c) \( 7x – y = 6 \)

Question. Let \( P = (1, 1) \) and \( Q = (3, 2) \). The point R on the x-axis such that \( PR + RQ \) is the minimum is
(a) \( \left( \frac{5}{3}, 0 \right) \)
(b) \( \left( \frac{1}{3}, 0 \right) \)
(c) \( (3, 0) \)
(d) none of the options
Answer: (a) \( \left( \frac{5}{3}, 0 \right) \)

Question. If a ray travelling along the \( x = 1 \) gets reflected from the line \( x + y = 1 \) then the equation of the line along which the reflected ray travels is
(a) \( y = 0 \)
(b) \( x – y = 1 \)
(c) \( x = 0 \)
(d) none of the options
Answer: (a) \( y = 0 \)

Question. The point \( P(2, 1) \) is shifted by \( 3\sqrt{2} \) parallel to the line \( x + y = 1 \), in the direction of increasing ordinate, to reach Q. The image of Q by the line \( x = y = 1 \) is
(a) \( (5, -2) \)
(b) \( (-1, 4) \)
(c) \( (3, -4) \)
(d) \( (-3, 2) \)
Answer: (d) \( (-3, 2) \)

Question. Let \( A = (1, 0) \) and \( B = (2, 1) \). The line AB turns about A through an angle \( \pi/6 \) in the clockwise sense, and the new position of B is B'. Then B' has the coordinates
(a) \( \left( \frac{3 + \sqrt{3}}{2}, \frac{\sqrt{3} - 1}{2} \right) \)
(b) \( \left( \frac{3 - \sqrt{3}}{2}, \frac{\sqrt{3} + 1}{2} \right) \)
(c) \( \left( \frac{1 - \sqrt{3}}{2}, \frac{1 + \sqrt{3}}{2} \right) \)
(d) none of the options
Answer: (a) \( \left( \frac{3 + \sqrt{3}}{2}, \frac{\sqrt{3} - 1}{2} \right) \)

Question. A line has intercepts \( a, b \) on the coordinate axes. If the axes are rotated about the origin through an angle \( \alpha \) then the line has intercepts \( p, q \) on the new position of the axes respectively. Then
(a) \( \frac{1}{p^2} + \frac{1}{q^2} = \frac{1}{a^2} + \frac{1}{b^2} \)
(b) \( \frac{1}{p^2} - \frac{1}{q^2} = \frac{1}{a^2} - \frac{1}{b^2} \)
(c) \( \frac{1}{p^2} + \frac{1}{a^2} = \frac{1}{q^2} + \frac{1}{b^2} \)
(d) none of the options
Answer: (a) \( \frac{1}{p^2} + \frac{1}{q^2} = \frac{1}{a^2} + \frac{1}{b^2} \)

Question. Two points A and B move on the x-axis and the y-axis respectively such that the distance between the two points is always the same. The locus of the middle point of AB is
(a) a straight line
(b) a pair of straight line
(c) a circle
(d) none of the options
Answer: (c) a circle

Question. Three vertices of a quadrilateral in order are \( (6, 1) \), \( (7, 2) \) and \( (-1, 0) \). If the area of the quadrilateral is 4 unit\( ^2 \) then the locus of the fourth vertex has the equation
(a) \( x – 7y = 1 \)
(b) \( x – 7y + 15 = 0 \)
(c) \( (x – 7y)^2 + 14(x – 7y) – 15 = 0 \)
(d) none of the options
Answer: (c) \( (x – 7y)^2 + 14(x – 7y) – 15 = 0 \)

Question. A variable line through the point \( (a, b) \) cuts the axes of reference at A and B respectively. The lines through A and B parallel to the y-axis and the x-axis respectively meet at P. Then the locus of P has the equation
(a) \( \frac{x}{a} + \frac{y}{b} = 1 \)
(b) \( \frac{x}{b} + \frac{y}{a} = 1 \)
(c) \( \frac{a}{x} + \frac{b}{y} = 1 \)
(d) \( \frac{b}{x} + \frac{a}{y} = 1 \)
Answer: (c) \( \frac{a}{x} + \frac{b}{y} = 1 \)

Choose the correct options. One or more options may be correct.

Question. If one vertex of an equilateral triangle of side 2 is the original and another vertex lies on the line \( x = \sqrt{3}y \) then the third vertex can be
(a) \( (0, 2) \)
(b) \( (\sqrt{3}, -1) \)
(c) \( (0, -2) \)
(d) \( (\sqrt{3}, 1) \)
Answer: (a) \( (0, 2) \), (b) \( (\sqrt{3}, -1) \)

Question. A line passing through the point \( (2, 2) \) and the axes enclose an area \( \lambda \). The intercepts on the axes made by the line are given by the two roots of
(a) \( x^2 – 2|\lambda|x + |\lambda| = 0 \)
(b) \( x^2 + |\lambda|x + 2|\lambda| = 0 \)
(c) \( x^2 - |\lambda|x + 2|\lambda| = 0 \)
(d) none of the options
Answer: (c) \( x^2 - |\lambda|x + 2|\lambda| = 0 \)

Question. A line passing through the origin and making an angle \( \pi/4 \) with the line \( y – 3x = 5 \) has the equation
(a) \( x + 2y = 0 \)
(b) \( 2x = y \)
(c) \( x = 2y \)
(d) \( y + 2x = 0 \)
Answer: (c) \( x = 2y \), (d) \( y + 2x = 0 \)

Question. The coordinates of a point on the line \( x + y = 3 \) such that the point is at equation distance from the line \( |x| = |y| \) are
(a) \( (3, 0) \)
(b) \( (0, 3) \)
(c) \( (-3, 0) \)
(d) \( (0, -3) \)
Answer: (a) \( (3, 0) \), (b) \( (0, 3) \)

Question. A line perpendicular to the line \( 3x – 2y = 5 \) cuts off an intercept 3 on the positive side of the x-axis. Then
(a) the slope of the line is \( \frac{2}{3} \)
(b) the intercept on the y-axis is 2
(c) the area of the triangle formed by the line with the axes is 3 unit\( ^2 \)
(d) none of the options
Answer: (b) the intercept on the y-axis is 2, (c) the area of the triangle formed by the line with the axes is 3 unit\( ^2 \)

Question. One diagonal of a square is the portion of the line \( \sqrt{3}x + y = 2\sqrt{3} \) intercepted by the axes. Then an extremity of the other diagonal is
(a) \( (1 + \sqrt{3}, \sqrt{3} - 1) \)
(b) \( (1 + \sqrt{3}, \sqrt{3} + 1) \)
(c) \( (1 - \sqrt{3}, \sqrt{3} - 1) \)
(d) \( (1 - \sqrt{3}, \sqrt{3} + 1) \)
Answer: (b) \( (1 + \sqrt{3}, \sqrt{3} + 1) \), (c) \( (1 - \sqrt{3}, \sqrt{3} - 1) \)

Question. If \( bx + cy = a \), where \( a, b, c \) are of the same sign, be a line such that the area enclosed by the line and the axes of reference is \( \frac{1}{8} \) unit\( ^2 \) then
(a) \( b, a, c \) are in GP
(b) \( a, 2a, c \) are in GP
(c) \( b, \frac{a}{2}, c \) are in AP
(d) \( b, -2a, c \) are in GP
Answer: (b) \( a, 2a, c \) are in GP, (d) \( b, -2a, c \) are in GP

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) Incentre
(d) Orthocentre
Answer: (b) Centroid, (c) Incentre

Question. If \( (x, y) \) be a variable point on the line \( y = 2x \) lying between the lines \( 2(x + 1) + y = 0 \) and \( x + 3(y – 1) = 0 \) then
(a) \( x \in \left( -\frac{1}{2}, \frac{6}{7} \right) \)
(b) \( x \in \left( -\frac{1}{2}, \frac{3}{7} \right) \)
(c) \( y \in \left( -1, \frac{3}{7} \right) \)
(d) \( y \in \left( -1, \frac{6}{7} \right) \)
Answer: (b) \( x \in \left( -\frac{1}{2}, \frac{3}{7} \right) \), (d) \( y \in \left( -1, \frac{6}{7} \right) \)

Question. If the equations of the three sides of a triangle are \( x + y = 1 \), \( 3x + 5y = 2 \) and \( x – y = 0 \) then the orthocenter of the triangle lies on the line
(a) \( 5x – 3y = 2 \)
(b) \( 3x – 5y + 1 = 0 \)
(c) \( 2x – 3y = 1 \)
(d) \( 5x – 3y = 1 \)
Answer: (b) \( 3x – 5y + 1 = 0 \), (d) \( 5x – 3y = 1 \)

Question. A ray travelling along the line \( 3x – 4y = 5 \) after being reflected from a line \( l \) travels along the line \( 5x + 12y = 13 \). Then the equation of the line \( l \) is
(a) \( x + 8y = 0 \)
(b) \( x = 8y \)
(c) \( 32x + 4y = 65 \)
(d) \( 32x – 4y + 65 = 0 \)
Answer: (b) \( x = 8y \), (c) \( 32x + 4y = 65 \)

Question. A ray of light travelling along the line \( x + y = 1 \) is incident on the x-axis and after refraction it enters the other side of the x-axis by turning \( \pi/6 \) away from the x-axis. The equation of the line along which the refracted ray travels is
(a) \( x + (2 - \sqrt{3})y = 1 \)
(b) \( (2 - \sqrt{3})x + y = 1 \)
(c) \( y + (2 + \sqrt{3})x = 2 + \sqrt{3} \)
(d) none of the options
Answer: (a) \( x + (2 - \sqrt{3})y = 1 \), (c) \( y + (2 + \sqrt{3})x = 2 + \sqrt{3} \)