Straight Lines PYQs Set 9 (PDF)

Set 9 of 15 · Questions 241-270 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 9 (Q241-270)
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241. $If the locus of the point, whose distances from the point (2, 1) and (1, 3) are in the ratio 5 : 4, is a x^{2} + b y^{2} + cxy + dx + ey + 170 = 0, then the value of a^{2} + 2 b + 3 c + 4 d + e is equal to :$ (1) $37$ (2) $437$ (3) $-27$ (4) $5$ 242. $If t \in R - -1, then the locus of the point (\frac{3 at}{1 + t ^{3}}, \frac{3 a t ^{2}}{1 + t ^{3}}) is$ (1) $x^{3} + y^{3} = 3 a x^{2} y^{2}$ (2) $x^{3} - 3 a x^{2} y - 3 ax y^{2} + y^{3} = 0$ (3) $x^{3} + y^{3} = 3 axy$ (4) $x^{3} - y^{3} = 3 axy$ 243. $For α \in (0, \frac{π}{2}), the angle between the lines represented by [x cos θ - y] [(cos θ + tan α) x - (1 - cos θ \cdot tan α) y] = 0 is$ (1) $α$ (2) $θ$ (3) $θ + α$ (4) $θ - α$ 244. $The locus of the point whose ratio of distance from the origin to its distance from (- 2, - 3) is 5 : 7, is given by\dots\dots\dots..$ (1) $24 x^{2} + 24 y^{2} - 100 x - 150 y - 325 = 0$ (2) $24 x^{2} + 24 y^{2} + 100 x + 150 y - 325 = 0$ (3) $24 x^{2} + 24 y^{2} - 100 x + 150 y + 325 = 0$ (4) $2 x^{2} + 2 y^{2} = 325$ 245. $A line L has intercepts a and b on the coordinate axes. When the axes are rotated through a given angle θ keeping the origin fixed, this line L has the intercepts p and q . Then$ (1) $a^{2} + b^{2} = p^{2} + q^{2}$ (2) $a^{2} + p^{2} = b^{2} + q^{2}$ (3) $\frac{1}{a ^{2}} + \frac{1}{p ^{2}} = \frac{1}{b ^{2}} + \frac{1}{q ^{2}}$ (4) $\frac{1}{a ^{2}} + \frac{1}{b ^{2}} = \frac{1}{p ^{2}} + \frac{1}{q ^{2}}$ 246. $The points A (1, 2), B (2, 4) and C (4, 8) form a/an$ (1) $isosceles triangle$ (2) $equilateral triangle$ (3) $straight line$ (4) $right angled triangle$ 247. $If h^{2} = ab, then the slopes of lines represented by a x^{2} - 2 hxy + b y^{2} = 0 would be in the ratio$ (1) $1 : 2$ (2) $2 : 1$ (3) $2 : 3$ (4) $1 : 1$ 248. $Consider the set of all lines px + qy + r = 0 such that 3 p + 2 q + 4 r = 0 . Which one of the following statements is true?$ (1) $The lines are concurrent at the point (\frac{3}{4}, \frac{1}{2})$ (2) $Each line passes through the origin.$ (3) $The lines are all parallel.$ (4) $The lines are not concurrent.$ 249. $If the straight, 2 x - 3 y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, β), then β equals :$ (1) $\frac{35}{3}$ (2) $- 5$ (3) $- \frac{35}{3}$ (4) $5$ 250. $If a x^{2} + 6 xy + b y^{2} - 10 x + 10 y - 6 = 0 represents a pair of perpendicular lines, then the values of \| a \| equals$ (1) $6$ (2) $4$ (3) $2$ (4) $3$ 251. $If the equation, x^{2} + 22 xy + 2 y^{2} + 4 x + 42 y + 1 = 0 represents a pair of straight lines which are parallel to each other. Find the distance between them is$ (1) $4 units$ (2) $2 units$ (3) $23 units$ (4) $43 units$ 252. $If the image of the point (- 4, 5) in the line x + 2 y = 2 lies on the circle (x + 4)^{2} + (y - 3)^{2} = r^{2}, then r is equal to:$ (1) $2$ (2) $3$ (3) $1$ (4) $4$ 253. $In △ ABC, if A is (1, 2), B and C lie on y = x + α (where α is variable), then the locus of the orthocenter of the triangle is$ (1) $x + y - 3 = 0$ (2) $x + y + 3 = 0$ (3) $y = x + 1$ (4) $y = x - 1$ 254. $Let R be a rectangle given by the lines x = 0, x = 2, y = 0 and y = 5. Let A(α, 0) and B(0, β), α \in [0, 2] and β \in [0, 5], be such that the line segment AB divides the area of the rectangle R in the ratio 4:1. Then, the mid-point of AB lies on a$ (1) $straight line$ (2) $parabola$ (3) $hyperbola$ (4) $circle$ 255. $Let A(l, 1), B(-4, 3) C(-2, -5) be vertices of a triangle ABC, P be a point on side BC, and ∆_{1} and ∆_{2} be the areas of triangle APB and ABC. Respectively. If ∆_{1} : ∆_{2} = 4:7, then the area enclosed by the lines AP, AC and the x-axis is$ (1) $\frac{1}{4}$ (2) $\frac{3}{4}$ (3) $\frac{1}{2}$ (4) $1$ 256. $If M is the foot of the perpendicular drawn from the origin O on to the variable line L, passing through a fixed point (a, b), then the locus of the mid-point of OM is$ (1) $x^{2} + y^{2} = a^{2} + b^{2}$ (2) $2 x^{2} + 2 y^{2} - ax - by = 0$ (3) $ax + by = 0$ (4) $2 x^{2} + 2 y^{2} - ay - bx = 0$ 257. $The combined equation of the diagonals of the square formed by the pairs of lines xy + 6 y - 4 x - 24 = 0 and xy + 6 x - 4 y - 24 = 0 is$ (1) $x^{2} - y^{2} + 2 x - 2 y = 0$ (2) $x^{2} - y^{2} + 2 x + 2 y = 0$ (3) $x^{2} - y^{2} + x + y = 0$ (4) $x^{2} + y^{2} - 2 xy + x - y = 0$ 258. $If A(2, -3) and B(-2, 1) are two vertices of a triangle and third vertex moves on the line 2x + 3y = 9 , then the locus of the centroid of the triangle is$ (1) $x - y = 1$ (2) $2x + 3y = 1$ (3) $2x + 3y = 3$ (4) $2x - 3y = 1$ 259. $A straight line makes an intercept on the Y -axis twice as long as that on X -axis and is at unit distance from the origin. Then the line is represented by the equations$ (1) $2 x + 3 y = \pm 5$ (2) $x + y = \pm 2$ (3) $x+y=\pm3$ (4) $2 x + y = \pm 5$ 260. $A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is \frac{1}{4} . Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then which of these stones is/are on the path of the man?$ (1) $B only$ (2) $A only$ (3) $All the three$ (4) $C only$ 261. $Two sides of a rhombus are along the lines, x - y + 1 = 0 and 7x - y - 5 = 0. if its diagonals intersect at (-1, -2), then which one of the following is a vertex of this rhombus ?$ (1) $(-3, -8)$ (2) $(\frac{1}{3}, - \frac{8}{3})$ (3) $(- \frac{10}{3}, - \frac{7}{3})$ (4) $(-3, -9)$ 262. $If the lines x^{2} - 4 xy + y^{2} = 0 and x + y = 10 contain the sides of an equilatetal triangle, then the area of equilateral triangle is$ (1) $\frac{5 2}{3} sq. units$ (2) $\frac{25 2}{3} sq. units$ (3) $\frac{50}{3} sq. units$ (4) $\frac{25}{3} sq. units$ 263. $If P (6, 1) be the orthocentre of the triangle whose vertices are A (5, - 2), B (8, 3) and C (h, k), then the point C lies on the circle:$ (1) $x^{2} + y^{2} - 61 = 0$ (2) $x^{2} + y^{2} - 52 = 0$ (3) $x^{2} + y^{2} - 65 = 0$ (4) $x^{2} + y^{2} - 74 = 0$ 264. $The combined equation of the two diameters of a circle which divide the circle into four sectors is a x^{2} + 2 hxy + b y^{2} = 0 . If the area of the bigger sector is 5 times the area of the smaller sector, then \frac{\| a + b \|}{( a - b ) ^{2} + 4 h ^{2}} is equal to$ (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $\frac{1}{2}$ (4) $1$ 265. $The lines x + y = \| a \| and ax - y = 1 intersect each other in the first quadrant. Then the set of all possible values of a is the interval :$ (1) $(0, \infty)$ (2) $[1, \infty)$ (3) $(-1, \infty)$ (4) $(-1, 1]$ 266. $Let A(0,1), -6(1,1) and (7(1,0) be the mid-points of the sides of a triangle with incentre at the point D. If the focus of the parabola y^{2} = 4ax passing through D is (α + β 2, 0), where α and β are rational numbers, then \frac{α ^{2}}{β ^{2}} is equal to$ (1) $8$ (2) $12$ (3) $6$ (4) $\frac{9}{2}$ 267. $N (3, - 4) is the foot of the perpendicular drawn from the origin to a line L . Then the equation of the line L is$ (1) $3 x - 4 y - 25 = 0$ (2) $4 x + 3 y = 0$ (3) $x - y - 7 = 0$ (4) $4 x - 3 y - 24 = 0$ 268. $Two equal sides of an isosceles triangle are along - x + 2 y = 4 and x + y = 4 . If m is the slope of its third side, then the sum, of all possible distinct values of m, is:$ (1) $- 210$ (2) $12$ (3) $6$ (4) $-6$ 269. $Let A (1, 0), B (6, 2) and C (\frac{3}{2}, 6) be the vertices of a triangle ABC . If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point (- \frac{7}{6}, - \frac{1}{3}), is \dots$ 270. $Consider a triangle ABC having the vertices A (1, 2), B (α, β) and C (γ, δ) and angles ∠ ABC = \frac{π}{6} and ∠ BAC = \frac{2 π}{3} . If the points B and C lie on the line y = x + 4, then α^{2} + γ^{2} is equal to _______$