JEE Mathematics Straight Lines MCQs Set 06

Question. The points \(\left(0, \frac{8}{3}\right)\), \((1,3)\) and \((82,30)\) are vertices of
(a) an obtuse angled triangle
(b) an acute angled triangle
(c) a right angled triangle
(d) None of the options
Answer: (d) None of the options

Question. The ratio in which the line joining the points \((3, -4)\) and \((-5, 6)\) is divided by x-axis
(a) 2 : 3
(b) 6 : 4
(c) 3 : 2
(d) None of the options
Answer: (a) 2 : 3
Solution: Let the ra

Question. The circumcentre of the triangle with vertices \((0, 0)\), \((3, 0)\) and \((0, 4)\) is
(a) \((1, 1)\)
(b) \((2, 3/2)\)
(c) \((3/2, 2)\)
(d) None of the options
Answer: (c) (3/2, 2)

Question. The mid points of the sides of a triangle are \((5, 0)\), \((5, 12)\) and \((0, 12)\), then orthocentre of this triangle is
(a) \((0, 0)\)
(b) \((0, 24)\)
(c) \((10, 0)\)
(d) \(\left(\frac{13}{3}, 8\right)\)
Answer: (a) (0, 0)

Question. Area of a triangle whose vertices are \((a \cos \theta, b \sin \theta)\), \((-a \sin \theta, b \cos \theta)\) and \((-a \cos \theta, -b \sin \theta)\) is
(a) \(ab \sin \theta \cos \theta\)
(b) \(a \cos \theta \sin \theta\)
(c) \(\frac{1}{2}ab\)
(d) \(ab\)
Answer: (d) ab

Question. The point A divides the join of the points \((-5, 1)\) and \((3, 5)\) in the ratio k : 1 and coordinates of points B and C are \((1, 5)\) and \((7, -2)\) respectively. If the area of \(\Delta ABC\) be 2 units, then k equals
(a) 7, 9
(b) 6, 7
(c) 7, 31/9
(d) 9, 31/9
Answer: (c) 7, 31/9

Question. If \(A(\cos\alpha, \sin\alpha)\), \(B(\sin\alpha, -\cos\alpha)\), \(C(1, 2)\) are the vertices of a \(\Delta ABC\), then as \(\alpha\) varies, the locus of its centroid is
(a) \(x^2 + y^2 - 2x - 4y + 3 = 0\)
(b) \(x^2 + y^2 - 2x - 4y + 1 = 0\)
(c) \(3(x^2 + y^2) - 2x - 4y + 1 = 0\)
(d) None of the options
Answer: (c) 3(x^2 + y^2) - 2x - 4y + 1 = 0

Question. The points with the co-ordinates \((2a, 3a)\), \((3b, 2b)\) & \((c, c)\) are collinear
(a) for no value of a, b, c
(b) for all values of a, b, c
(c) If a, c/5, b are in H.P.
(d) if a, 2c/5, b are in H.P.
Answer: (d) if a, 2c/5, b are in H.P.

Question. A stick of length 10 units rests against the floor and a wall of a room. If the stick begins to slide on the floor then the locus of its middle point is
(a) \(x^2 + y^2 = 2.5\)
(b) \(x^2 + y^2 = 25\)
(c) \(x^2 + y^2 = 100\)
(d) None of the options
Answer: (b) x^2 + y^2 = 25

Question. The equation of the line cutting an intercept of 3 on negative y-axis and inclined at an angle \(\tan^{-1}\frac{3}{5}\) to the x-axis is
(a) \(5y - 3x + 15 = 0\)
(b) \(5y - 3x = 15\)
(c) \(3y - 5x + 15 = 0\)
(d) None of the options
Answer: (a) 5y - 3x + 15 = 0

Question. The equation of a straight line which passes through the point \((-3, 5)\) such that the portion of it between the axes is divided by the point in the ratio 5 : 3 (reckoning from x-axis) will be
(a) \(x + y - 2 = 0\)
(b) \(2x + y + 1 = 0\)
(c) \(x + 2y - 7 = 0\)
(d) \(x - y + 8 = 0\)
Answer: (d) x - y + 8 = 0

Question. The co-ordinates of the vertices P, Q, R & S of square PQRS inscribed in the triangle ABC with vertices \(A(0, 0)\), \(B(3, 0)\) & \(C(2, 1)\) given that two of its vertices P, Q are on the side AB are respectively
(a) \(\left(\frac{1}{8}, 0\right)\), \(\left(\frac{3}{8}, 0\right)\), \(\left(\frac{3}{8}, \frac{1}{4}\right)\) & \(\left(\frac{1}{8}, \frac{1}{4}\right)\)
(b) \(\left(\frac{1}{4}, 0\right)\), \(\left(\frac{3}{4}, 0\right)\), \(\left(\frac{3}{4}, \frac{1}{2}\right)\) & \(\left(\frac{1}{4}, \frac{1}{2}\right)\)
(c) \((1, 0)\), \(\left(\frac{3}{2}, 0\right)\), \(\left(\frac{3}{2}, \frac{1}{2}\right)\) & \(\left(1, \frac{1}{2}\right)\)
(d) \(\left(\frac{3}{2}, 0\right)\), \(\left(\frac{9}{4}, 0\right)\), \(\left(\frac{9}{4}, \frac{3}{4}\right)\) & \(\left(\frac{3}{2}, \frac{3}{4}\right)\)
Answer: (d) \(\left(\frac{3}{2}, 0\right)\), \(\left(\frac{9}{4}, 0\right)\), \(\left(\frac{9}{4}, \frac{3}{4}\right)\) & \(\left(\frac{3}{2}, \frac{3}{4}\right)\)

Question. The equation of perpendicular bisector of the line segment joining the points \((1, 2)\) and \((-2, 0)\) is
(a) \(5x + 2y = 1\)
(b) \(4x + 6y = 1\)
(c) \(6x + 4y = 1\)
(d) None of the options
Answer: (c) 6x + 4y = 1

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

Question. Points A & B are in the first quadrant ; point 'O' is the origin. If the slope of OA is 1, slope of OB is 7 and OA = OB, then the slope of AB is
(a) \(-1/5\)
(b) \(-1/4\)
(c) \(1/3\)
(d) \(-1/2\)
Answer: (d) -1/2

Question. Coordinates of a point which is at 3 distance from point \((1, -3)\) of line \(2x + 3y + 7 = 0\) is
(a) \(\left(1 + \frac{9}{\sqrt{13}}, -3 - \frac{6}{\sqrt{13}}\right)\)
(b) \(\left(1 \mp \frac{9}{\sqrt{13}}, -3 \pm \frac{6}{\sqrt{13}}\right)\)
(c) \(\left(1 + \frac{9}{\sqrt{13}}, -3 + \frac{6}{\sqrt{13}}\right)\)
(d) \(\left(1 - \frac{9}{\sqrt{13}}, 3 - \frac{6}{\sqrt{13}}\right)\)
Answer: (b) \(\left(1 \mp \frac{9}{\sqrt{13}}, -3 \pm \frac{6}{\sqrt{13}}\right)\)

Question. The angle between the lines \(y - x + 5 = 0\) and \(\sqrt{3}x - y + 7 = 0\) is
(a) \(15^\circ\)
(b) \(60^\circ\)
(c) \(45^\circ\)
(d) \(75^\circ\)
Answer: (a) 15°

Question. A line is perpendicular to \(3x + y = 3\) and passes through a point \((2, 2)\). Its y intercept is
(a) \(2/3\)
(b) \(1/3\)
(c) 1
(d) \(4/3\)
Answer: (d) 4/3

Question. The equation of the line passing through the point \((c, d)\) and parallel to the line \(ax + by + c = 0\) is
(a) \(a(x + c) + b(y + d) = 0\)
(b) \(a(x + c) - b(y + d) = 0\)
(c) \(a(x - c) + b(y - d) = 0\)
(d) None of the options
Answer: (c) a(x - c) + b(y - d) = 0

Question. The position of the point \((8, -9)\) with respect to the lines \(2x + 3y - 4 = 0\) and \(6x + 9y + 8 = 0\) is
(a) point lies on the same side of the lines
(b) point lies on one of the lines
(c) point lies on the different sides of the line
(d) None of the options
Answer: (a) point lies on the same side of the lines

Question. If origin and \((3, 2)\) are contained in the same angle of the lines \(2x + y - a = 0\), \(x - 3y + a = 0\), then 'a' must lie in the interval
(a) \((-\infty, 0) \cup (8, \infty)\)
(b) \((-\infty, 0) \cup (3, \infty)\)
(c) \((0, 3)\)
(d) \((3, 8)\)
Answer: (a) (-∞, 0) U (8, ∞)

Question. The line \(3x + 2y = 6\) will divide the quadrilateral formed by the lines \(x + y = 5\), \(y - 2x = 8\), \(3y + 2x = 0\) & \(4y - x = 0\) in
(a) two quadrilaterals
(b) one pentagon and one triangle
(c) two triangles
(d) None of the options
Answer: (a) two quadrilaterals

Question. If the point \((a, 2)\) lies between the lines \(x - y - 1 = 0\) and \(2(x - y) - 5 = 0\), then the set of values of a is
(a) \((-\infty, 3) \cup (9/2, \infty)\)
(b) \((3, 9/2)\)
(c) \((-\infty, 3)\)
(d) \((9/2, \infty)\)
Answer: (b) (3, 9/2)

Question. \(A(x_1, y_1)\), \(B(x_2, y_2)\) and \(C(x_3, y_3)\) are three non-collinear points in cartesian plane. Number of parallelograms that can be drawn with these three points as vertices are
(a) one
(b) two
(c) three
(d) four
Answer: (c) three

Question. If \(P(1, 0)\) ; \(Q(-1, 0)\) & \(R(2, 0)\) are three given points, then the locus of the points S satisfying the relation, \(SQ^2 + SR^2 = 2 SP^2\) is
(a) A straight line parallel to x-axis
(b) A circle passing through the origin
(c) A circle with the centre at the origin
(d) A straight line parallel to y-axis
Answer: (d) A straight line parallel to y-axis

Question. The area of triangle formed by the lines \(x + y - 3 = 0\), \(x - 3y + 9 = 0\) and \(3x - 2y + 1 = 0\)
(a) 16/7 sq. units
(b) 10/7 sq. units
(c) 4 sq. units
(d) 9 sq. units
Answer: (b) 10/7 sq. units

Question. The co-ordinates of foot of the perpendicular drawn on line \(3x - 4y - 5 = 0\) from the point \((0, 5)\) is
(a) \((1, 3)\)
(b) \((2, 3)\)
(c) \((3, 2)\)
(d) \((3, 1)\)
Answer: (d) (3, 1)

Question. Distance of the point \((2, 5)\) from the line \(3x + y + 4 = 0\) measured parallel to the line \(3x - 4y + 8 = 0\) is
(a) 15/2
(b) 9/2
(c) 5
(d) None of the options
Answer: (c) 5

Question. Three vertices of triangle ABC are \(A(-1, 11)\), \(B(-9, -8)\) and \(C(15, -2)\). The equation of angle bisector of angle A is
(a) \(4x - y = 7\)
(b) \(4x + y = 7\)
(c) \(x + 4y = 7\)
(d) \(x - 4y = 7\)
Answer: (b) 4x + y = 7

Question. If line \(y - x + 2 = 0\) is shifted parallel to itself towards the positive direction of the x-axis by a perpendicular distance of \(3\sqrt{2}\) units, then the equation of the new line is
(a) \(y = x - 4\)
(b) \(y = x + 1\)
(c) \(y = x - (2 + 3\sqrt{2})\)
(d) \(y = x - 8\)
Answer: (d) y = x - 8

Question. The co-ordinates of the point of reflection of the origin \((0, 0)\) in the line \(4x - 2y - 5 = 0\) is
(a) \((1, -2)\)
(b) \((2, -1)\)
(c) \(\left(\frac{2}{5}, \frac{4}{5}\right)\)
(d) \((2, 5)\)
Answer: (b) (2, -1)

Question. If the axes are rotated through an angle of \(30^\circ\) in the anti-clockwise direction, the coordinates of point \((4, -2\sqrt{3})\) with respect to new axes are
(a) \((2, \sqrt{3})\)
(b) \((\sqrt{3}, -5)\)
(c) \((2, 3)\)
(d) \((\sqrt{3}, 2)\)
Answer: (b) (\sqrt{3}, -5)

Question. Keeping the origin constant axes are rotated at an angle \(30^\circ\) in clockwise direction then new coordinate of \((2, 1)\) with respect to old axes is
(a) \(\left(\frac{2\sqrt{3}}{2}, \frac{2}{2\sqrt{3}}\right)\)
(b) \(\left(\frac{2\sqrt{3} + 1}{2}, \frac{2 - \sqrt{3}}{2}\right)\)
(c) \(\left(\frac{2\sqrt{3} - 2}{2}, \frac{2\sqrt{3} + 1}{2}\right)\)
(d) None of the options
Answer: (b) \(\left(\frac{2\sqrt{3}+1}{2}, \frac{2-\sqrt{3}}{2}\right)\)

Question. If one diagonal of a square is along the line \(x = 2y\) and one of its vertex is \((3, 0)\), then its sides through this vertex are given by the equations
(a) \(y - 3x + 9 = 0\), \(x - 3y - 3 = 0\)
(b) \(y - 3x + 9 = 0\), \(x - 3y - 3 = 0\)
(c) \(y + 3x - 9 = 0\), \(x + 3y - 3 = 0\)
(d) \(y - 3x + 9 = 0\), \(x + 3y - 3 = 0\)
Answer: (d) y - 3x + 9 = 0, x + 3y - 3 = 0

Question. The line \((p + 2q)x + (p - 3q)y = p - q\) for different values of p and q passes through a fixed point whose co-ordinates are
(a) \(\left(\frac{3}{2}, \frac{5}{2}\right)\)
(b) \(\left(\frac{2}{5}, \frac{2}{5}\right)\)
(c) \(\left(\frac{3}{5}, \frac{3}{5}\right)\)
(d) \(\left(\frac{2}{5}, \frac{3}{5}\right)\)
Answer: (d) (2/5, 3/5)

Question. Given the family of lines, \(a(3x+4y+6) + b(x+y+2)=0\). The line of the family situated at the greatest distance from the point \(P(2, 3)\) has equation
(a) \(4x + 3y + 8 = 0\)
(b) \(5x + 3y + 10 = 0\)
(c) \(15x + 8y + 30 = 0\)
(d) None of the options
Answer: (a) 4x + 3y + 8 = 0

Question. The base BC of a triangle ABC is bisected at the point \((p, q)\) and the equation to the side AB & AC are \(px + qy = 1\) & \(qx + py = 1\). The equation of the median through A is
(a) \((p - 2q)x + (q - 2p)y + 1 = 0\)
(b) \((p + q)x + y - 2 = 0\)
(c) \((2pq - 1)(px + qy - 1) = (p^2 + q^2 - 1)(qx + py - 1)\)
(d) None of the options
Answer: (c) (2pq - 1)(px + qy - 1) = (p^2 + q^2 - 1)(qx + py - 1)

Question. The equation \(2x^2 + 4xy - py^2 + 4x + qy + 1 = 0\) will represent two mutually perpendicular straight lines, if
(a) \(p = 1\) and \(q = 2\) or \(6\)
(b) \(p = -2\) and \(q = -2\) or \(8\)
(c) \(p = 2\) and \(q = 0\) or \(8\)
(d) \(p = 2\) and \(q = 0\) or \(6\)
Answer: (c) p = 2 and q = 0 or 8

Question. Equation of the pair of straight lines through origin and perpendicular to the pair of straight lines \(5x^2 - 7xy - 3y^2 = 0\) is
(a) \(3x^2 - 7xy - 5y^2 = 0\)
(b) \(3x^2 + 7xy + 5y^2 = 0\)
(c) \(3x^2 - 7xy + 5y^2 = 0\)
(d) \(3x^2 + 7xy - 5y^2 = 0\)
Answer: (a) 3x^2 - 7xy - 5y^2 = 0

Question. One of the diameter of the circle circumscribing the rectangle ABCD is \(4y = x + 7\). If A and B are the points \((-3, 4)\) and \((5, 4)\) respectively then the area of rectangle is equal to
(a) 30
(b) 8
(c) 25
(d) 32
Answer: (d) 32