JEE Mathematics Straight Lines MCQs Set 07

Question. If the lines \(x\sin^2A + y\sin A + 1 = 0\), \(x\sin^2B + y\sin B + 1 = 0\), \(x\sin^2C + y\sin C + 1 = 0\) are concurrent where A, B, C are angles of triangle then \(\Delta ABC\) must be
(a) equilateral
(b) isosceles
(c) right angle
(d) no such triangle exist
Answer: (b) isosceles

Question. The co-ordinates of a point P on the line \(2x - y + 5 = 0\) such that \(|PA - PB|\) is maximum where A is \((4, -2)\) and B is \((2, -4)\) will be
(a) \((11, 27)\)
(b) \((-11, -17)\)
(c) \((-11, 17)\)
(d) \((0, 5)\)
Answer: (b) (-11, -17)

Question. The line \(x + y = p\) meets the axis of x and y at A and B respectively. A triangle APQ is inscribed in the triangle OAB, O being the origin, with right angle at Q, P and Q lie respectively on OB and AB. If the area of the triangle APQ is \(3/8^{th}\) of the area of the triangle OAB, then AQ/BQ is equal to
(a) 2
(b) 2/3
(c) 1/3
(d) 3
Answer: (d) 3

Question. Lines, \(L_1 : x + \sqrt{3}y = 2\), and \(L_2 : ax + by = 1\), meet at P and enclose an angle of \(45^\circ\) between them. Line \(L_3 : y = \sqrt{3}x\), also passes through P then
(a) \(a^2 + b^2 = 1\)
(b) \(a^2 + b^2 = 2\)
(c) \(a^2 + b^2 = 3\)
(d) \(a^2 + b^2 = 4\)
Answer: (b) a^2 + b^2 = 2

Question. A triangle is formed by the lines \(2x - 3y - 6 = 0\) ; \(3x - y + 3 = 0\) and \(3x + 4y - 12 = 0\). If the points \(P(\alpha, 0)\) and \(Q(0, \beta)\) always lie on or inside the \(\Delta ABC\), then
(a) \(\alpha \in [-1, 2]\) & \(\beta \in [-2, 3]\)
(b) \(\alpha \in [-1, 3]\) & \(\beta \in [-2, 4]\)
(c) \(\alpha \in [-2, 4]\) & \(\beta \in [-3, 4]\)
(d) \(\alpha \in [-1, 3]\) & \(\beta \in [-2, 3]\)
Answer: (d) \(\alpha \in [-1, 3]\) & \(\beta \in [-2, 3]\)

Question. The line \(x + 3y - 2 = 0\) bisects the angle between a pair of straight lines of which one has equation \(x - 7y + 5 = 0\). The equation of the other line is
(a) \(3x + 3y - 1 = 0\)
(b) \(x - 3y + 2 = 0\)
(c) \(5x + 5y - 3 = 0\)
(d) None of the options
Answer: (c) 5x + 5y - 3 = 0

Question. A ray of light passing through the point A(1, 2) is reflected at a point B on the x-axis and then passes through (5, 3). Then the equation of AB is
(a) \(5x + 4y = 13\)
(b) \(5x - 4y = -3\)
(c) \(4x + 5y = 14\)
(d) \(4x - 5y = -6\)
Answer: (a) 5x + 4y = 13

Question. Let the algebraic sum of the perpendicular distances from the point (3, 0), (0, 3) & (2, 2) to a variable straight line be zero, then the line passes through a fixed point whose co-ordinates are
(a) \((3, 2)\)
(b) \((2, 3)\)
(c) \(\left(\frac{3}{5}, \frac{3}{5}\right)\)
(d) \(\left(\frac{5}{3}, \frac{5}{3}\right)\)
Answer: (d) (5/3, 5/3)

Question. The image of the pair of lines represented by \(ax^2 + 2h xy + by^2 = 0\) by the line mirror y = 0 is
(a) \(ax^2 - 2hxy + by^2 = 0\)
(b) \(bx^2 - 2hxy + ay^2 = 0\)
(c) \(bx^2 + 2hxy + ay^2 = 0\)
(d) \(ax^2 - 2hxy - by^2 = 0\)
Answer: (a) ax^2 - 2hxy + by^2 = 0

Question. The pair of straight lines \(x^2 - 4xy + y^2 = 0\) together with the line \(x + y + 4\sqrt{6} = 0\) form a triangle which is
(a) right angled but not isosceles
(b) right isosceles
(c) scalene
(d) equilateral
Answer: (d) equilateral

Question. Let \(A \equiv (3, 2)\) and \(B \equiv (5, 1)\). ABP is an equilateral triangle is constructed on the side of AB remote from the origin then the orthocentre of triangle ABP is
(a) \(\left(4 - \frac{1}{2}\sqrt{3}, \frac{3}{2} - \sqrt{3}\right)\)
(b) \(\left(4 + \frac{1}{2}\sqrt{3}, \frac{3}{2} + \sqrt{3}\right)\)
(c) \(\left(4 - \frac{1}{6}\sqrt{3}, \frac{3}{2} - \frac{1}{3}\sqrt{3}\right)\)
(d) \(\left(4 + \frac{1}{6}\sqrt{3}, \frac{3}{2} + \frac{1}{3}\sqrt{3}\right)\)
Answer: (d) \(\left(4 + \frac{1}{6}\sqrt{3}, \frac{3}{2} + \frac{1}{3}\sqrt{3}\right)\)

Question. The line PQ whose equation is \(x - y = 2\) cuts the x-axis at P and Q is \((4, 2)\). The line PQ is rotated about P through \(45^\circ\) in the anticlockwise direction. The equation of the line PQ in the new position is
(a) \(y = -\sqrt{2}\)
(b) \(y = 2\)
(c) \(x = 2\)
(d) \(x = -2\)
Answer: (c) x = 2

Question. Distance between two lines represented by the line pair, \(x^2 - 4xy + 4y^2 + x - 2y - 6 = 0\) is
(a) \(1/\sqrt{5}\)
(b) \(\sqrt{5}\)
(c) \(2\sqrt{5}\)
(d) None of the options
Answer: (b) \(\sqrt{5}\)

Question. The circumcentre of the triangle formed by the lines, \(xy + 2x + 2y + 4 = 0\) and \(x + y + 2 = 0\) is
(a) \((-1, -1)\)
(b) \((-2, -2)\)
(c) \((0, 0)\)
(d) \((-1, -2)\)
Answer: (a) (-1, -1)

Question. Area of the rhombus bounded by the four lines, \(ax \pm by \pm c = 0\) is
(a) \(\frac{c^2}{2ab}\)
(b) \(\frac{2c^2}{|ab|}\)
(c) \(\frac{4c^2}{ab}\)
(d) \(\frac{ab}{4c^2}\)
Answer: (b) \(\frac{2c^2}{|ab|}\)

Question. If the lines \(ax + y + 1 = 0\), \(x + by + 1 = 0\) & \(x + y + c = 0\) where a, b & c are distinct real numbers different from 1 are concurrent, then the value of \(\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c}\) equals
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (d) 1

Question. The area enclosed by \(2 | x | + 3| y | \le 6\) is
(a) 3 sq. units
(b) 4 sq. units
(c) 12 sq. units
(d) 24 sq. units
Answer: (c) 12 sq. units

Question. The point \((4, 1)\) undergoes the following three transformations successively
(i) Reflection about the line \(y = x\)
(ii) Translation through a distance 2 units along the positive direction of x-axis
(iii) Rotation through an angle \(\pi/4\) about the origin in the counter clockwise direction.
The final position of the points is given by the coordinates
(a) \(\left(\frac{7}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)\)
(b) \(\left(\frac{7}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)
(c) \(\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\)
(d) None of the options
Answer: (c) \(\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\)

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

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

Question. If \( \frac{x}{c} + \frac{y}{d} = 1 \) is a line through the intersection of \( \frac{x}{a} + \frac{y}{b} = 1 \) and \( \frac{x}{b} + \frac{y}{a} = 1 \) and the lengths of the perpendiculars drawn from the origin to these lines are equal in lengths then
(a) \( \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2} + \frac{1}{d^2} \)
(b) \( \frac{1}{a^2} - \frac{1}{b^2} = \frac{1}{c^2} - \frac{1}{d^2} \)
(c) \( \frac{1}{a} + \frac{1}{b} = \frac{1}{c} + \frac{1}{d} \)
(d) None of the options
Answer: (a) \( \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2} + \frac{1}{d^2} \), (c) \( \frac{1}{a} + \frac{1}{b} = \frac{1}{c} + \frac{1}{d} \)

Question. A and B are two fixed points whose co-ordinates are (3, 2) and (5, 4) respectively. The co-ordinates of a point P if ABP is an equilateral triangle, is/are
(a) \( (4 - \sqrt{3}, 3 + \sqrt{3}) \)
(b) \( (4 + \sqrt{3}, 3 - \sqrt{3}) \)
(c) \( (3 - \sqrt{3}, 4 + \sqrt{3}) \)
(d) \( (3 + \sqrt{3}, 4 - \sqrt{3}) \)
Answer: (a) \( (4 - \sqrt{3}, 3 + \sqrt{3}) \), (b) \( (4 + \sqrt{3}, 3 - \sqrt{3}) \)

Question. Straight lines \( 2x + y = 5 \) and \( x - 2y = 3 \) intersect at the point A. Points B and C are chosen on these two lines such that AB = AC. Then the equation of a line BC passing through the point (2, 3) is
(a) \( 3x - y - 3 = 0 \)
(b) \( x + 3y - 11 = 0 \)
(c) \( 3x + y - 9 = 0 \)
(d) \( x - 3y + 7 = 0 \)
Answer: (a) 3x - y - 3 = 0, (b) x + 3y - 11 = 0

Question. The straight lines \( x + y = 0 \), \( 3x + y - 4 = 0 \) and \( x + 3y - 4 = 0 \) form a triangle which is
(a) isosceles
(b) right angled
(c) obtuse angled
(d) equilateral
Answer: (a) isosceles, (c) obtuse angled