Straight Lines PYQs Set 4 (PDF)

Set 4 of 15 · Questions 91-120 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 4 (Q91-120)
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91. $Suppose P (x, y) lyingon 3 x - y + 2 = 0 or 3 x + y - 2 = 0 is at a distance of 5 units from their point of intersection. Then the distance from (0, 0) to the foot of the perpendicular of P onto the y -axis is$ (1) $2 + \frac{5 3}{2}$ (2) $\frac{5 3}{2}$ (3) $2$ (4) $2 - \frac{5 3}{2}$ 92. $Let a variable line of slope m > 0 passing through the point (4, - 9) intersect the coordinate axes at the points A and B . The minimum value of the sum of the distances of A and B from the origin is$ (1) $30$ (2) $25$ (3) $15$ (4) $10$ 93. $If slopes of lines represented by K x^{2} + 5 xy + y^{2} = 0 differ by 1 , then K =$ (1) $2$ (2) $3$ (3) $6$ (4) $8$ 94. $If the sum of the slopes of the lines given by x^{2} - 2 cxy - 11 y^{2} = 0 is six times their product, then c has the value$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 95. $The equation y^{2} - x^{2} + 6 x - 9 = 0 represents$ (1) $a pair of straight lines$ (2) $a circle$ (3) $a parabola$ (4) $an ellipse$ 96. $Line has slope m and y -intercept 4 . The distance between the origin and the line is equal to$ (1) $\frac{4}{1 - m ^{2}}$ (2) $\frac{4}{m ^{2} - 1}$ (3) $\frac{4}{m ^{2} + 1}$ (4) $\frac{4 m}{m ^{2} + 1}$ 97. $If A (4, 0) and B (- 4, 0) are two points, then the locus of a point P such that PA - PB = 4 is$ (1) $3 x^{2} - y^{2} = 12$ (2) $x^{2} - 3 y^{2} = 12$ (3) $4 (x^{2} - 3 y^{2}) = 1$ (4) $3 x^{2} - y^{2} = 1$ 98. $If m_{1} and m_{2}, (m_{1} > m_{2}) are the slopes of the lines represented by 5 x^{2} - 8 xy + 3 y^{2} = 0, then m_{1} : m_{2} equals$ (1) $5 : 1$ (2) $2 : 1$ (3) $5 : 3$ (4) $3 : 2$ 99. $The two lines lx + my = n and l^{’} x + m^{’} y = n^{’} are perpendicular if$ (1) $l m^{’} = m l^{’}$ (2) $lm + l^{’} m^{’} = 0$ (3) $l m^{’} + m l^{’} = 0$ (4) $l l^{’} + m m^{’} = 0$ 100. $The combined equation of the two lines ax + by + c = 0 and a^{’} x + b^{’} y + c^{’} = 0 can be written as (ax + by + c) (a^{’} x + b^{’} y + c^{’}) = 0 the equation of the angle bisectors of the lines represented by the equation 2 x^{2} + xy - 3 y^{2} = 0 is$ (1) $3 x^{2} + 5 xy + 2 y^{2} = 0$ (2) $x^{2} - y^{2} + 10 y^{2} = 0$ (3) $3 x^{2} - xy - 2 y^{2} = 0$ (4) $x^{2} - y^{2} - 10 xy = 0$ 101. $If θ is the acute angle between the pair of lines 12 x^{2} + 2 hxy + 7 y^{2} = 0 and tan θ = \frac{8}{19}, then h =$ (1) $\pm 6$ (2) $\pm 7$ (3) $\pm 8$ (4) $\pm 10$ 102. $Find the equation of pair of straight lines that bisect the angles between the lines represented by a x^{2} + 2 hxy + b y^{2} = 0 .$ (1) $\frac{x ^{2} + y ^{2}}{a + b} = \frac{xy}{h}$ (2) $\frac{x ^{2} + y ^{2}}{a - b} = \frac{xy}{h}$ (3) $\frac{x ^{2} + y ^{2}}{a - b} = \frac{h}{xy}$ (4) $\frac{x ^{2} - y ^{2}}{a - b} = \frac{xy}{h}$ 103. $For a, b, c \in R, if 6 a^{2} - 3 b^{2} - c^{2} + 7 ab - ac + 4 bc = 0 and \| a \| + \| b \| \neq = 0, then all the lines given by ax + by + c = 0 are$ (1) $concurrent at (3, 1) or (1, 3)$ (2) $parallel to each other \forall a, b, c \in R$ (3) $concurrent at (- 2, - 3) or (3, - 1)$ (4) $concurrent at (2, 3) or (- 3, 1)$ 104. $If α x^{2} + 2 γxy + β y^{2} = 0 is the equation of pair of lines passing through the origin and perpendicular to the pair of lines bh x^{2} + abxy + ah y^{2} = 0 (a \neq = 0, b \neq = 0), then \frac{αβ}{γ ^{2}} =$ (1) $\frac{h ^{2}}{ab}$ (2) $\frac{- 2 h ^{2}}{ab}$ (3) $\frac{- h ^{2}}{ab}$ (4) $\frac{4 h ^{2}}{ab}$ 105. $Let P = (- 1, 0), Q = (0, 0) and R \equiv (3, 33) be three points. The equation of the bisector of the angle PQR is$ (1) $3 x - y = 0$ (2) $x - 3 y = 0$ (3) $3 x + y = 0$ (4) $x + 3 y = 0$ 106. $A \equiv (cos θ, sin θ), B \equiv (sin θ, - cos θ) are two points. The locus of the centroid of △ OAB, where ' O' is the origin is$ (1) $x^{2} + y^{2} = 3$ (2) $2 x^{2} + 2 y^{2} = 9$ (3) $9 x^{2} + 9 y^{2} = 2$ (4) $3 x^{2} + 3 y^{2} = 2$ 107. $Find the equation of a straight line passing through (- 5, 6) and cutting off equal intercepts on the coordinate axes.$ (1) $6 x - 5 y = 30$ (2) $x - y = - 11$ (3) $x + y = 11$ (4) $x + y = 1$ 108. $The line which is parallel to X -axis and crosses the curve y = x at an angle of 45^{\circ} is$ (1) $y = \frac{1}{4}$ (2) $y = \frac{1}{2}$ (3) $y = 1$ (4) $y = 4$ 109. $If the lines 3 x + 4 y - 5 = 0, 2 x + 3 y - 4 = 0 and px + 4 y - 6 = 0 all meet at the same point, therr p is equal to$ (1) $-2$ (2) $0$ (3) $1$ (4) $2$ 110. $B (2, 3), C (5, - 2) . D (1, - 1) are three points. If A is a variable point such that the area of the quadrilateral ABCD is 10 sq. units, then the locus of A is$ (1) $(x - 4 y + 42) (x - 4 y + 2) = 0$ (2) $(x - 4 y - 42) (x - 4 y - 2) = 0$ (3) $(4 x - y + 42) (4 x - y + 2) = 0$ (4) $(4 x - y - 42) (4 x - y - 2) = 0$ 111. $If the line 3 x + 4 y - 24 = 0 intersects the x -axis at the point A and the y -axis at the point B, then the incentre of the triangle OAB, where O is the origin is:$ (1) $(3, 4)$ (2) $(2, 2)$ (3) $(4, 3)$ (4) $(4, 4)$ 112. $The set of all possible values of θ in the interval (0, π) for which the points (1, 2) and (sin θ, cos θ) lie on the same side of the line x + y = 1 is :$ (1) $(0, \frac{π}{2})$ (2) $(\frac{π}{4}, \frac{3 π}{4})$ (3) $(0, \frac{π}{4})$ (4) $(0, \frac{3 π}{4})$ 113. $Equation of the locus of the centroid of the triangle whose vertices are (a cos k, a sin k), (b sin k, - b cos k) and (1, 0), where k is a parameter, is$ (1) $(1 - 3 x)^{2} + 9 y^{2} = a^{2} + b^{2}$ (2) $(3 x - 1)^{2} + 9 y^{2} = 2 a^{2} + 2 b^{2}$ (3) $(3 x + 1)^{2} + (3 y)^{2} = a^{2} + b^{2}$ (4) $(3 x + 1)^{2} + (3 y)^{2} = 3 a^{2} + 3 b^{2}$ 114. $The length of the perpendicular from the origin on the normal to the curve, x^{2} + 2 xy - 3 y^{2} = 0 at the point (2, 2) is:$ (1) $42$ (2) $2$ (3) $22$ (4) $4$ 115. $If x^{2} - y^{2} + 2 hxy + 2 gx + 2 fy + c = 0 is the locus of a point, which moves such that it is always equidistant from the lines x + 2 y + 7 = 0 and 2 x - y + 8 = 0, then the value of g + c + h - f equals$ (1) $14$ (2) $6$ (3) $8$ (4) $29$ 116. $When the coordinate axes are rotated about the origin in the positive direction through an angle \frac{π}{4}, if the equation 25 x^{2} + 9 y^{2} = 225 is transformed to α x^{2} + βxy + γ y^{2} = δ, then (α + β + γ - δ)^{2} =$ (1) $3$ (2) $9$ (3) $4$ (4) $16$ 117. $The equation of the line through the intersection of 3 x - 4 y + 1 = 0 and 5 x + y - 1 = 0 which cuts off equal intercepts on the axes is given by$ (1) $23 x + 23 y - 11 = 0$ (2) $23 x + 23 y + 11 = 0$ (3) $23 x - 23 y - 11 = 0$ (4) $23 x - 23 y + 11 = 0$ 118. $If A (- 3, 3), B (1, 1), C (1, - 1) and D (- 2, - 2) are the vertices of a quadrilateral, the angle between the diagonals AC and BD is$ (1) $\frac{π}{4}$ (2) $\frac{π}{2}$ (3) $\frac{π}{6}$ (4) $\frac{π}{3}$ 119. $When the origin is shifted to the point (\frac{3}{2}, \frac{3}{2}) by the translation of coordinate axes, then the transformed equation of 32 x^{2} + 8 xy + 32 y^{2} - 108 x - 108 y + 99 = 0 is$ (1) $72 X^{2} + 56 Y^{2} - 63 = 0$ (2) $X^{2} - 14 XY - 7 Y^{2} - 2 = 0$ (3) $32 X^{2} - 16 XY + 32 Y^{2} - 225 = 0$ (4) $32 X^{2} + 8 XY + 32 Y^{2} - 63 = 0$ 120. $Let a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point (- 5, 0) . If the locus of the point P is a circle of radius r, then 4 r^{2} is equal to (Round off to the nearest integer)$