If the lines 3x – 4y – 7 = 0 and 2x – 3y – 5 = 0 are two diameters of a circle of area square units, then what will be the equation of the circle?
In an ellipse, the distance between its focii is 6 and the minor axis is 8. Then its eccentricity is,
A straight line through the point A(3, 4) is such that its intercept between the axes is bisected at A. Its equation is,
If the pair of lines ax2 + 2(a + b)xy + by2 = 0 lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then
An ellipse has OB as semi-minor axis, F and F' its focii and the angle FBF' is a right angle. Then what will be the eccentricity of the ellipse?
If a circle passes through the point (a, b) and cuts the circle x2 + y2 = p2 orthogonally, then the equation of the locus of its centre is
A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. What will be the locus of the centre of the circle?
A particle of mass 4 m which is at rest explodes into three fragments, two of the fragments each of mass m are found to move each with a speed v making an angle 90 with each other. The total energy relased in this explosion is -
A slow moving electron collides elastically with a hydrogen atom at rest. The initial and final motions are along the same straight line. What fraction of electron's kinetic energy is transferred to the hydrogen atom? The mass of hydrogen atom is 1850 times the mass of electron?
If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 – 3ax + dy – 1 = 0 intersect in two distinct points P and Q then the line 5x+by – a = 0 passes through P and Q for
If a vertex of a triangle is (1, 1) and the midpoints of two sides through this vertex are (–1, 2) and (3, 2), then what will be the centroid of the triangle.
If non zero numbers a, b, c are in H.P., then the straight line always passes through a fixed point. That point is
The line parallel to thexaxis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx – 2ay – 3a = 0, where (a, b) (0, 0) is
The mass of a rocket is 500 kg and the relative velocity of the gases ejecting from it is 250 m/s with respect to the rocket. The rate of burning of the fuel in order to give the rocket an initial acceleration 20 m/s2 in the vertically upward direction (g = 10 m/s2), will be
A 20 g bullet pierces through a plate of mass m1 = 1 kg and then comes to rest inside a second plate of mass m2 = 2.98 kg. It is found that the two plates, initially at rest, now move with equal velocities. The percentage loss in the initial velocity of bullet when it is between m1 and m2. (Neglect any loss of material of the bodies, due to action of bullet.) will be
If the straight lines x = 1 + s, y = –3 z = 1 + and x = t/2, y = 1 + t, z = 2 – t, with parameters s and t respectively, are coplanar, then equals
A bullet of mass 10g travelling horizontally with a velocity of 300 m/s strikes a block of wood of mass 290 g which rests on a rough horizontal floor. After impact the block and the bullet move together and come to rest when the block has travelled a distance of 15 m. The coefficient of friction between the block and the floor will be - (Duration of impact is very short)
Calculate empirical formula weight from the given values and molecular weight from freezing point depression?
The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directrices is x = 4, then the equation of the ellipse is
The intercept on the line y = x by the circle x2 + y2 – 2x = 0 is AB. Equation of the circle on AB as diameter is
If the lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 lie along diameters of a circle of circumference then the equation of the circle is
If a 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas y2 = 4ax and x2 = 4ay, then
A variable circle passes through the fixed point A(p, q) and touches x-axis. The locus of the other end of the diameter through A is
If a circle passes through the point (a, b) and cuts the circle x2 + y2 = 4 orthogonally, then the locus of its centre is
If the sum of the slopes of the lines given by x2 – 2cxy – 7y2 = 0 is four times their product, then c has the value
The equation of the straight line passing through the point (4, 3) and making intercepts on the co ordinate axes whose sum is –1 is
Let A(2, –3) and B(–2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line
A point on the parabola y2 = 18x at which the ordinate increases at twice the rate of the abscissa is
The centres of a set of circles, each of radius 3, lie on the circle x2 + y2 = 25. The locus of any point in the set is
Locus of the centroid of the triangle whose vertices are (a cost, a sint), (b sint, –b cost) and (1, 0), where t is a parameter, is
A rifle man,who together with his rifle has a mass of 100 kg, stands on a smooth surface fires 10 shots horizontally. Each bullet has a mass 10 gm and a muzzle velocity of 800 m/s. What velocity does rifle man acquire at the end of 10 shots
A uniform rod of length 4 m and mass 20 kg is lying horizontal on the ground. The work done in keeping it vertical with one of its ends touching the ground, will be
If the pairs of straight lines x2 – 2pxy – y2 = 0 and x2 – 2qxy – y2 = 0 be such that each pair bisects the angle between the other pair, then
The work done by a person in carrying a box of mass 10 kg through a vertical height of 10 m is 4900 J. The mass of the person is
A particle of mass m is moving in a horizontal circle of radius r, under a centripetal force equal to (–k/r2), where k
is a constant. The total energy of the particle is
A bus of mass 1000 kg has an engine which produces a constant power of 50 kW. If the resistance to motion, assumed constant is 1000 N. The maximum speed at which the bus can travel on level road and the acceleration when it is travelling at 25 m/s, will respectively be?
A uniform chain is held on a frictionless table with one- fifth of its length hanging over the edge. If the chain has a length l and a mass m, how much work is required to pull the hanging part back on the table ?
A boy pulls a 5 kg block 20 metres along a horizontal sur- face at a constant speed with a force directed 45° above the horizontal. If the coefficient of kinetic friction is 0.20, how much work does the boy do on the block?
A square of side a lies above the x axis and has one vertex at the origin. The side passing through the origin makes an angle with the positive direction of the x-axis. The equation of its diagonal not passing through the origin is
The lines 2x – 3y= 5 and 3x – 4y= 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is
If the two circles (x – 1)2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0 intersect in two distinct points, then
The normal at the point (bt12, 2bt1) on a parabola meets the parabola again in the point (bt22, 2bt2), then