JEE Mathematics Coordinates and Straight Lines MCQs Set 01

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

Question. If two vertices of an equilateral triangle have integral coordinates then the third vertex will have
(a) integral coordinates
(b) coordinates which are rational
(c) at least one coordinate irrational
(d) coordinates which are irrational
Answer: (c) at least one coordinate irrational

Question. If the line segment joining \( (2, 3) \) and \( (-1, 2) \) is divided internally in the ration \( 3 : 4 \) by the line \( x + 2y = k \) then \( k \) is
(a) \( \frac{41}{7} \)
(b) \( \frac{5}{7} \)
(c) \( \frac{36}{7} \)
(d) \( \frac{31}{7} \)
Answer: (a) \( \frac{41}{7} \)

Question. The polar coordinates of the vertices of a triangle are \( (0, 0) \), \( (3, \pi/2) \) and \( (3, \pi/6) \). Then the triangle is
(a) right angled
(b) isosceles
(c) equilateral
(d) none of the options
Answer: (c) equilateral

Question. The point \( (a, b + c) \), \( (b, c + a) \), \( (c, a + b) \) are
(a) vertices of an equilateral triangle
(b) collinear
(c) concyclic
(d) none of the options
Answer: (b) collinear

Question. The incentre of the triangle formed by the axes and the line \( \frac{x}{a} + \frac{y}{b} = 1 \) is
(a) \( \left( \frac{a}{2}, \frac{b}{2} \right) \)
(b) \( \left( \frac{ab}{a + b + \sqrt{ab}}, \frac{ab}{a + b + \sqrt{ab}} \right) \)
(c) \( \left( \frac{a}{3}, \frac{b}{3} \right) \)
(d) \( \left( \frac{ab}{a + b + \sqrt{a^2 + b^2}}, \frac{ab}{a + b + \sqrt{a^2 + b^2}} \right) \)
Answer: (d) \( \left( \frac{ab}{a + b + \sqrt{a^2 + b^2}}, \frac{ab}{a + b + \sqrt{a^2 + b^2}} \right) \)

Question. In the \( \Delta ABC \), the coordinates of B are \( (0, 0) \), \( AB = 2 \), \( \angle ABC = \frac{\pi}{3} \) and the middle point of BC has the coordinates \( (2, 0) \). The centroid of the triangle is
(a) \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \)
(b) \( \left( \frac{5}{3}, \frac{1}{\sqrt{3}} \right) \)
(c) \( \left( \frac{4 + \sqrt{3}}{3}, \frac{1}{3} \right) \)
(d) none of the options
Answer: (b) \( \left( \frac{5}{3}, \frac{1}{\sqrt{3}} \right) \)

Question. The coordinates of three consecutive vertices of a parallelogram are \( (1, 3) \), \( (-1, 2) \) and \( (2, 5) \). The coordinates of the fourth vertex are
(a) \( (6, 4) \)
(b) \( (4, 6) \)
(c) \( (-2, 0) \)
(d) none of the options
Answer: (b) \( (4, 6) \)

Question. The area of the pentagon whose vertices are \( (4, 1) \), \( (3, 6) \), \( (-5, 1) \), \( (-3, -3) \) and \( (-3, 0) \) is
(a) 30 unit\( ^2 \)
(b) 60 unit\( ^2 \)
(c) 120 unit\( ^2 \)
(d) none of the options
Answer: (a) 30 unit\( ^2 \)

Question. A point moves in the x-y plane such that the sum of its distances from two mutually perpendicular lines is always equal to 3. The area enclosed by the locus of the point is
(a) 18 unit\( ^2 \)
(b) \( \frac{9}{2} \) unit\( ^2 \)
(c) 9 unit\( ^2 \)
(d) none of the options
Answer: (a) 18 unit\( ^2 \)

Question. Let \( A = (1, 2) \), \( B = (3, 4) \) and let \( C = (x, y) \) be a point such that \( (x – 1)(x – 3) + (y – 2)(y – 4) = 0 \). If \( ar (\Delta ABC) = 1 \) then maximum number of positions of C in the x-y plane is
(a) 2
(b) 4
(c) 8
(d) none of the options
Answer: (b) 4

Question. The points \( (\alpha, \beta) \), \( (\gamma, \delta) \), \( (\alpha, \delta) \) and \( (\gamma, \beta) \) taken in order, where \( \alpha, \beta, \gamma, \delta \) are different real numbers, are
(a) collinear
(b) vertices of a square
(c) vertices of a rhombus
(d) concyclic
Answer: (d) concyclic

Question. The diagonals of a parallelogram PQRS are along the lines \( x + 3y = 4 \) and \( 6x – 2y = 7 \). The PQRS must be a
(a) rectangle
(b) square
(c) cyclic quadrilateral
(d) rhombus
Answer: (d) rhombus

Question. The coordinates of the four vertices of a quadrilateral are \( (-2, 4) \), \( (-1, 2) \), \( (1, 2) \) and \( (2, 4) \) taken in order. The equation of the line passing through the vertex \( (-1, 2) \) and dividing the quadrilateral in two equal areas is
(a) \( x + 1 = 0 \)
(b) \( x + y = 1 \)
(c) \( x – y + 3 = 0 \)
(d) none of the options
Answer: (c) \( x – y + 3 = 0 \)

Question. The equation of the straight line which passes through the point \( (-4, 3) \) such that the portion of the line between the axes is divided internally by the point in the ratio \( 5 : 3 \) is
(a) \( 9x – 20y + 96 = 0 \)
(b) \( 9x + 20y = 24 \)
(c) \( 20x + 9y + 53 = 0 \)
(d) none of the options
Answer: (a) \( 9x – 20y + 96 = 0 \)

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

Question. The equation of a straight line passing through the point \( (-2, 3) \) and making intercepts of equal length on the axes is
(a) \( 2x + y + 1 = 0 \)
(b) \( x – y = 5 \)
(c) \( x – y + 5 = 0 \)
(d) none of the options
Answer: (c) \( x – y + 5 = 0 \)

Question. The foot of the perpendicular on the line \( 3x + y = \lambda \) drawn from the origin is C. If the line cuts the x-axis and y-axis at A and B respectively then \( BC : CA \) is
(a) \( 1 : 3 \)
(b) \( 3 : 1 \)
(c) \( 1 : 9 \)
(d) \( 9 : 1 \)
Answer: (d) \( 9 : 1 \)

Question. The distance of the line \( 2x – 3y = 4 \) from the point \( (1, 1) \) in the direction of the lien \( x + y = 1 \) is
(a) \( \sqrt{2} \)
(b) \( 5\sqrt{2} \)
(c) \( \frac{1}{\sqrt{2}} \)
(d) none of the options
Answer: (a) \( \sqrt{2} \)

Question. The four sides of a quadrilateral are given by the equation \( xy(x – 2)(y – 3) = 0 \). The equation of the line parallel to \( x – 4y = 0 \) that divides the quadrilateral in two equal areas is
(a) \( x – 4y + 5 = 0 \)
(b) \( x – 4y – 5 = 0 \)
(c) \( 4y = x + 1 \)
(d) \( 4y + 1 = x \)
Answer: (a) \( x – 4y + 5 = 0 \)

Question. The coordinates of two consecutive vertices A and B of a regular hexagon ABCDEF are \( (1, 0) \) and \( (2, 0) \) respectively. The equation of the diagonal CE is
(a) \( \sqrt{3}x + y = 4 \)
(b) \( x + \sqrt{3}y + 4 = 0 \)
(c) \( x + \sqrt{3}y = 4 \)
(d) none of the options
Answer: (c) \( x + \sqrt{3}y = 4 \)

Question. ABC is an isosceles triangle in which A is \( (-1, 0) \), \( \angle A = 2\pi/3 \), \( AB = AC \) and AB is along the x-axis. If \( BC = 4\sqrt{3} \) then the equation of the line BC is
(a) \( x + \sqrt{3}y = 3 \)
(b) \( \sqrt{3}x + y = 3 \)
(c) \( x + y = \sqrt{3} \)
(d) none of the options
Answer: (a) \( x + \sqrt{3}y = 3 \)

Question. The graph of the function \( \cos x \cdot \cos(x + 2) – \cos^2(x + 1) \) is a
(a) straight line passing through the point \( (0, -\sin^2 1) \) with slope 2
(b) straight line passing through the origin
(c) parabola with vertex \( (1, -\sin^2 1) \)
(d) straight line passing through the point \( (\pi/2, -\sin^2 1) \) and parallel to the x-axis
Answer: (d) straight line passing through the point \( (\pi/2, -\sin^2 1) \) and parallel to the x-axis

Question. If the points \( (-2, 0) \), \( (-1, 1/\sqrt{3}) \) and \( (\cos \theta, \sin \theta) \) are collinear then the number of value of \( \theta \in [0, 2\pi] \) is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (b) 1

Question. The limiting position of the point of intersection of the lines \( 3x + 4y = 1 \) and \( (1 + c)x + 3c^2y = 2 \) as c tends to 1 is
(a) \( (-5, 4) \)
(b) \( (5, -4) \)
(c) \( (4, -5) \)
(d) none of the options
Answer: (a) \( (-5, 4) \)

Question. The coordinate of the point on the x-axis which is equidistant from the points \( (-3, 4) \) and \( (2, 5) \) are
(a) \( (20, 0) \)
(b) \( (-23, 0) \)
(c) \( \left( \frac{4}{5}, 0 \right) \)
(d) none of the options
Answer: (d) none of the options

Question. The distance between the line \( 3x + 4y = 9 \) and \( 6x + 8y + 15 = 0 \) is
(a) \( \frac{3}{10} \)
(b) \( \frac{33}{10} \)
(c) \( \frac{33}{5} \)
(d) none of the options
Answer: (b) \( \frac{33}{10} \)

Question. If a vertex of an equilateral triangle is the origin and the side opposite to it has the equation \( x + y = 1 \) then the orthocenter of the triangle is
(a) \( \left( \frac{1}{3}, \frac{1}{3} \right) \)
(b) \( \left( \frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3} \right) \)
(c) \( \left( \frac{2}{3}, \frac{2}{3} \right) \)
(d) none of the options
Answer: (a) \( \left( \frac{1}{3}, \frac{1}{3} \right) \)

Question. The equation of the three sides of a triangle are \( x = 2 \), \( y + 1 = 0 \) and \( x + 2y = 4 \). The coordinates of the circumcentre of the triangle are
(a) \( (4, 0) \)
(b) \( (2, -1) \)
(c) \( (0, 4) \)
(d) none of the options
Answer: (a) \( (4, 0) \)

Question. L is a variable line such that the algebraic sum of the distances of the points \( (1, 1) \), \( (2, 0) \) and \( (0, 2) \) from the line is equal to zero. The line L will always pass through
(a) \( (1, 1) \)
(b) \( (2, 1) \)
(c) \( (1, 2) \)
(d) none of the options
Answer: (a) \( (1, 1) \)

Question. ABC is an equilateral triangle such that the vertices B and C lie on two parallel lines at a distance 6. If A lies between the parallel lines at a distance 4 from one of them the length of a side of the equilateral triangle is
(a) 8
(b) \( \sqrt{\frac{88}{3}} \)
(c) \( \frac{4\sqrt{7}}{\sqrt{3}} \)
(d) none of the options
Answer: (c) \( \frac{4\sqrt{7}}{\sqrt{3}} \)

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

Question. If the coordinates of the vertices of a triangle are rational numbers then which of the following points of the triangle will always have rational coordinates ?
(a) centroid
(b) Incentre
(c) Circumcentre
(d) Orthocentre
Answer: (a) centroid, (c) Circumcentre, (d) Orthocentre

Question. Two consecutive vertices of a rectangle of area 10 unit\( ^2 \) are \( (1, 3) \) and \( (-2, -1) \). Other two vertices are
(a) \( \left( -\frac{3}{5}, \frac{21}{5} \right), \left( -\frac{18}{5}, \frac{1}{5} \right) \)
(b) \( \left( -\frac{3}{5}, \frac{21}{5} \right), \left( -\frac{2}{5}, -\frac{11}{5} \right) \)
(c) \( \left( -\frac{2}{5}, -\frac{11}{5} \right), \left( \frac{13}{5}, \frac{9}{5} \right) \)
(d) \( \left( \frac{13}{5}, \frac{9}{5} \right), \left( \frac{18}{5}, \frac{1}{5} \right) \)
Answer: (a) \( \left( -\frac{3}{5}, \frac{21}{5} \right), \left( -\frac{18}{5}, \frac{1}{5} \right) \), (c) \( \left( -\frac{2}{5}, -\frac{11}{5} \right), \left( \frac{13}{5}, \frac{9}{5} \right) \)

Question. The ends of a diagonal of a square are \( (2, -3) \) and \( (-1, 1) \). Another vertex of the square can be
(a) \( \left( -\frac{3}{2}, -\frac{5}{2} \right) \)
(b) \( \left( \frac{5}{2}, \frac{1}{2} \right) \)
(c) \( \left( \frac{1}{2}, \frac{5}{2} \right) \)
(d) none of the options
Answer: (a) \( \left( -\frac{3}{2}, -\frac{5}{2} \right) \), (b) \( \left( \frac{5}{2}, \frac{1}{2} \right) \)

Question. If each of the vertices of a triangle has integral coordinates then the triangle may be
(a) right angled
(b) equilateral
(c) isosceles
(d) none of the options
Answer: (a) right angled, (c) isosceles, (d) none of the options

Question. If \( (-1, 2) \), \( (2, -1) \) and \( (3, 1) \) are any three vertices of a parallelogram then the fourth vertex \( (a, b) \) will be such that
(a) \( a = 2, b = 0 \)
(b) \( a = -2, b = 0 \)
(c) \( a = -2, b = 6 \)
(d) \( a = 6, b = -2 \)
Answer: (b) \( a = -2, b = 0 \), (d) \( a = 6, b = -2 \)

Question. If \( (\alpha, \beta) \) be an end of a diagonal of a square and the other diagonal has the equation \( x – y = \alpha \) then another vertex of the square can be
(a) \( (\alpha  \beta, \alpha) \)
(b) \( (\alpha, 0) \)
(c) \( (0, -\alpha) \)
(d) \( (\alpha + \beta, \beta) \)
Answer: (b) \( (\alpha, 0) \), (d) \( (\alpha + \beta, \beta) \)

Question. A point on the line \( y = x \) whose perpendicular distance from the line \( \frac{x}{4} + \frac{y}{3} = 1 \) is 4 has the coordinates
(a) \( \left( -\frac{8}{7}, -\frac{8}{7} \right) \)
(b) \( \left( \frac{32}{7}, \frac{32}{7} \right) \)
(c) \( \left( \frac{3}{2}, \frac{3}{2} \right) \)
(d) none of the options
Answer: (a) \( \left( -\frac{8}{7}, -\frac{8}{7} \right) \), (b) \( \left( \frac{32}{7}, \frac{32}{7} \right) \)

Question. The parametric equation of a line is given by \( x = -2 + \frac{r}{\sqrt{10}} \) and \( y = 1 + \frac{3r}{\sqrt{10}} \). Then, for the line
(a) intercept on the x-axis = \( \frac{7}{3} \)
(b) intercept on the y-axis = -7
(c) slope of the line = \( \tan^{-1} \frac{1}{3} \)
(d) slope of the line = \( \tan^{-1} 3 \)
Answer: (d) slope of the line = \( \tan^{-1} 3 \)

Question. One side of a square of length \( a \) is inclined to the x-axis at an angle \( \alpha \) with one of the vertices of the square at the origin. The equation of a diagonal of the square is
(a) \( y(\cos \alpha - \sin \alpha) = x(\cos \alpha + \sin \alpha) \)
(b) \( y(\cos \alpha + \sin \alpha) = x(\cos \alpha - \sin \alpha) \)
(c) \( y(\sin \alpha + \cos \alpha) – x(\sin \alpha - \cos \alpha) = a \)
(d) \( y(\sin \alpha + \cos \alpha) + x(\sin \alpha - \cos \alpha) = a \)
Answer: (a) \( y(\cos \alpha - \sin \alpha) = x(\cos \alpha + \sin \alpha) \), (c) \( y(\sin \alpha + \cos \alpha) – x(\sin \alpha - \cos \alpha) = a \)

Question. If the equations of the hypotenuse and a side of a right-angled isosceles triangles be \( x + my = 1 \) and \( x = k \) respectively then
(a) \( m = 1 \)
(b) \( m = k \)
(c) \( m = -1 \)
(d) \( m + k = 0 \)
Answer: (a) \( m = 1 \), (c) \( m = -1 \)

Question. The centroid and a vertex of an equilateral triangle are \( (1, 1) \) and \( (1, 2) \) respectively. Another vertex of the triangle can be
(a) \( \left( \frac{2 - \sqrt{3}}{2}, \frac{1}{2} \right) \)
(b) \( \left( \frac{2 + 3\sqrt{3}}{2}, \frac{1}{2} \right) \)
(c) \( \left( \frac{2 + \sqrt{3}}{2}, \frac{1}{2} \right) \)
(d) none of the options
Answer: (a) \( \left( \frac{2 - \sqrt{3}}{2}, \frac{1}{2} \right) \), (c) \( \left( \frac{2 + \sqrt{3}}{2}, \frac{1}{2} \right) \)