Straight Lines PYQs Set 6 (PDF)

Set 6 of 15 · Questions 151-180 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 6 (Q151-180)
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151. $If P (α, β) be a point on the line 3 x + y = 0 such that the point P and the point Q (1, 1) lie on either side of the line 3 x = 4 y + 8, then$ (1) $α > \frac{8}{15}, β < \frac{- 8}{5}$ (2) $α < \frac{8}{15}, β < \frac{- 8}{5}$ (3) $α > \frac{8}{15}, β > \frac{- 8}{5}$ (4) $α < \frac{8}{15}, β > \frac{- 8}{5}$ 152. $The points (11, 9), (2, 1) and (2, - 1) are the midpoints of the sides of the triangle. Then the centroid is$ (1) $(5, 3)$ (2) $(- 5, - 3)$ (3) $(5, - 3)$ (4) $(3, 5)$ 153. $A value of k such that the straight lines y - 3 kx + 4 = 0 and (2 k - 1) x - (8 k - 1) y - 6 = 0 are perpendicular is$ (1) $\frac{1}{6}$ (2) $- \frac{1}{6}$ (3) $1$ (4) $0$ 154. $Let the equation of the pair of lines, y = pxandy = qx, can be written as (y - px)(y - qx) = 0. Then the equation of the pair of the angle bisectors of the lines x^{2} - 4xy - 5 y^{2} = 0 is:$ (1) $x^{2} - 3xy + y^{2} = 0$ (2) $x^{2} + 4xy - y^{2} = 0$ (3) $x^{2} + 3xy - y^{2} = 0$ (4) $x^{2} - 3xy - y^{2} = 0$ 155. $Suppose a triangle of area 27 sq. units is formed by 18 x^{2} - 9 xy + y^{2} = 0 and y = c . Then the centroid of the triangle is$ (1) $(3, 12)$ (2) $(12, 3)$ (3) $(- 12, 3)$ (4) $(- 3, 12)$ 156. $Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4ax + 2ay + c = 0 and 5bx + 2by + d = 0 lies in the fourth quadrant and is equidistant from the two axes then :$ (1) $3bc - 2ad = 0$ (2) $3bc + 2ad = 0$ (3) $2bc - 3ad = 0$ (4) $2bc + 3ad = 0$ 157. $If θ is the acute angle between the pair of lines H \equiv a x^{2} - xy + b y^{2} = 0, tan θ = 5 and (1, - 1) is a point on H = 0 then a^{2} + ab + b^{2} =$ (1) $5$ (2) $14$ (3) $7$ (4) $13$ 158. $The point to which the origin is to be shifted to remove the first degree terms from the equation 2 x^{2} + 4 xy - 6 y^{2} + 2 x + 8 y + 1 = 0 is$ (1) $(\frac{7}{8}, \frac{3}{8})$ (2) $(\frac{- 7}{8}, \frac{- 3}{8})$ (3) $(\frac{- 7}{8}, \frac{3}{8})$ (4) $(\frac{7}{8}, \frac{- 3}{8})$ 159. $A line cuts off equal intercepts on the co-ordinate axes. The angle made by this line with the positive direction of X -axis is$ (1) $45^{\circ}$ (2) $120^{\circ}$ (3) $90^{\circ}$ (4) $135^{\circ}$ 160. $When the axis are rotated through an angle 45^{\circ}, the new coordinates of a point P are (1, - 1) . The coordinates of P in the original system are$ (1) $(2, 2)$ (2) $(2, 0)$ (3) $(0, 2)$ (4) $(- 2, 0)$ 161. $The point P (1, 4) occupies the positions A, B and C respectively after undergoing the following three transformation successively. I. Reflection about the line y = x . II. Translation through a distance of 1 unit along the positive direction of X -axis. III. Rotation of the line OB through an angle \frac{π}{4} about the origin in the anti-clockwise direction. Then, the coordinates of C are$ (1) $(2, 22)$ (2) $(22, 32)$ (3) $(\frac{5}{2}, \frac{7}{2})$ (4) $(\frac{2}{2}, \frac{3}{2})$ 162. $If the centroid of the triangle formed by the lines 2 y^{2} + 5 xy - 3 x^{2} = 0 and x + y = k is (\frac{1}{18}, \frac{11}{18}), then the value of k equals$ (1) $-1$ (2) $0$ (3) $1$ (4) $2$ 163. $If 3 x + 6 y + 2 = 0, x + y + 1 = 0, 2 x - y + 3 = 0 are three given lines then the point (\frac{- 4}{3}, \frac{1}{3}) is$ (1) $the orthocenter of the triangle formed by the lines$ (2) $the point of concurrence of the lines$ (3) $the circumcentre of the triangle formed by the lines$ (4) $the incentre of the triangle formed by the lines$ 164. $The angle between the straight lines x^{2} + 4 xy + y^{2} = 0 is\dots\dots\dots$ (1) $30^{\circ}$ (2) $45^{\circ}$ (3) $60^{\circ}$ (4) $90^{\circ}$ 165. $If the lines L_{1} \equiv x - 2 y + 3 = 0, L_{2} \equiv 2 x + y + 1 = 0 and L_{3} \equiv 3 x + y + c = 0 are concurrent and θ is the acute angle between the lines L_{1} = 0 and L_{3} = 0, then tan θ =$ (1) $c + 2$ (2) $c - 5$ (3) $c + 5$ (4) $c - 2$ 166. $The perpendiculars are drawn to lines L_{1} and L_{2} from the origin making an angle \frac{π}{4} and \frac{3 π}{4} respectively with positive direction of X -axis. If both the lines are at unit distance from the origin, then their joint equation is$ (1) $x^{2} - y^{2} + 22 y + 2 = 0$ (2) $x^{2} - y^{2} - 22 y - 2 = 0$ (3) $x^{2} - y^{2} + 22 y - 2 = 0$ (4) $x^{2} - y^{2} - 22 y + 2 = 0$ 167. $If the origin of a coordinate system is shifted to (- 2, 2) and the coordinate system is rotated anticlockwise through an angle 45^{\circ}, then the point P (1, - 1) in the original system has new coordinates$ (1) $(2, - 22)$ (2) $(0, - 22)$ (3) $(0, - 2 - 2)$ (4) $(0, - 2 + 2)$ 168. $The number of real values of α for which the pair of lines represented by (α^{2} + 12 \| α \|) x^{2} + 6 xy + (18 - 21 \| α \|) y^{2} = 0 are at right angles to each other, is$ (1) $0$ (2) $1$ (3) $2$ (4) $4$ 169. $If 2 x^{2} - 5 xy + 2 y^{2} = 0 represents two sides of a triangle whose centroid is (1, 1), then the equation of the third side is$ (1) $x + y + 3 = 0$ (2) $x - y - 3 = 0$ (3) $x + y - 3 = 0$ (4) $x - y + 3 = 0$ 170. $If lines represented by x + 3 y - 6 = 0, 2 x + y - 4 = 0 and kx - 3 y + 1 = 0 are concurrent, then the value of k is$ (1) $6/19$ (2) $19/6$ (3) $- 19/6$ (4) $- 6/19$ 171. $The family of lines, forming an isosceles triangle with the lines 3 x - 4 y - 2 = 0 and 12 x - 5 y + 6 = 0, is$ (1) $9 x + 7 y + c = 0$ (2) $7 x - 9 y + c = 0$ (3) $9 x - 7 y + c = 0$ (4) $x \pm y + c = 0$ 172. $If lines represented by equation p x^{2} - q y^{2} = 0 are distinct, then$ (1) $pq > 0$ (2) $pq < 0$ (3) $pq = 0$ (4) $p + q = 0$ 173. $The number of integral values of m so that the abscissa of point of intersection of lines 3 x + 4 y = 9 and y = m x + 1 is also an integer, is :$ (1) $1$ (2) $2$ (3) $3$ (4) $0$ 174. $The lines p(p2 + 1) x - y + q = 0 and (p2 + 1)2 x + (p2 + 1) y + 2q = 0 are perpendicular to a common line for:$ (1) $exactly one value of p$ (2) $exactly two values of p$ (3) $more than two values of p$ (4) $no value of p$ 175. $The equation of the pair of straight lines passing through the point (2, 3) and perpendicular to the pair of lines 3 x^{2} - 4 xy + 5 y^{2} = 0 is a x^{2} + 2 hxy + b y^{2} + 2 gx + 2 fy + c = 0 then a + b + c + f + g + h =$ (1) $0$ (2) $52$ (3) $25$ (4) $-54$ 176. $If the lines, joining the origin to the points of intersection of the curve 2 x^{2} - 2 xy + 3 y^{2} + 2 x - y - 1 = 0 and the line x + 2 y = k, are at right angles, then k^{2} equals$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 177. $The co-ordinates of the foot of the perpendicular drawn from the point (3, 4) on the line 2 x + y - 7 = 0 is$ (1) $(\frac{9}{5}, \frac{17}{5})$ (2) $(1, 5)$ (3) $(- 5, 1)$ (4) $(1, - 5)$ 178. $Lines are drawn parallel to the line 4 x - 3 y + 2 = 0, at a distance \frac{3}{5} from the origin. Then which one of the following points lies on any of these lines ?$ (1) $(- \frac{1}{4}, \frac{2}{3})$ (2) $(\frac{1}{4}, - \frac{1}{3})$ (3) $(\frac{1}{4}, \frac{1}{3})$ (4) $(- \frac{1}{4}, - \frac{2}{3})$ 179. $The distance from the origin to the image of (1, 1) with respect to the line x + y + 5 = 0 is$ (1) $72$ (2) $32$ (3) $62$ (4) $42$ 180. $Let tan α, tan β and tan γ; α, β, γ \neq = \frac{( 2 n - 1 ) π}{2}, n \in N be the slopes of three line segments O A, O B and O C, respectively, where O is origin. If circumcentre of Δ A B C coincides with origin and its orthocentre lies on y -axis, then the value of (\frac{c o s 3 α + c o s 3 β + c o s 3 γ}{c o s α c o s β c o s γ})^{2} is equal to :$