Chapter-5-1.CONIC SECTION-01- (THEORY)

5.0.1. Introduction. The curves obtained by intersection of a plane and a double cone in different orientation are called conic section. In other words “Graph of a quadratic equation (in two variables) is a “Conic section”. A conic section or conic is the locus of a point P, which moves in such a way that its distance from a fixed point S always bears a constant ratio to its distance from a fixed straight line, all being in the same plane. constant = e (eccentricity) or SP= e. PM 5.0.2. Definitions of Various important Terms. (1) Focus : The fixed point is called the focus of the conic-section. (2) Directrix : The fixed straight line is called the directrix of the conic section. In general, every central conic has four foci, two of which are real and the other two are imaginary. Due to two real foci, every conic has two directrices corresponding to each real focus. (3) Eccentricity : The constant ratio is called the eccentricity of the conic section and is denoted by e. If , the conic is called Parabola. If e < 1, the conic is called Ellipse. If , the conic is called Hyperbola. If , the conic is called Circle. If , the conic is called Pair of the straight lines. (4) Axis: The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section. A conic is always symmetric about its axis. (5) Vertex: The points of intersection of the conic section and the axis are called vertices of conic section. (6) Centre: The point which bisects every chord of the conic passing through it, is called the centre of conic. (7) Latus-rectum: The latus-rectum of a conic is the chord passing through the focus and perpendicular to the axis. (8) Double ordinate: The double ordinate of a conic is a chord perpendicular to the axis. (9) Focal chord: A chord passing through the focus of the conic is called a focal chord. (10) Focal distance: The distance of any point on the conic from the focus is called the focal distance of the point. 5.0.3. General equation of a Conic section when its Focus, Directrix and Eccentricity are given Let be the focus, be the directrix and e be the eccentricity of a conic. Let be any point on the conic. Let PM be the perpendicular from P, on the directrix. Then by definition   Thus the locus of is this is the cartesian equation of the conic section which, when simplified, can be written in the form , which is general equation of second degree. 5.0.4. Recognisation of Conics. The equation of conics is represented by the general equation of second degree ......(i) and discriminant of above equation is represented by , where Case I: When In this case equation (i) represents the degenerate conic whose nature is given in the following table. S. No. Condition Nature of conic 1. and A pair of coincident straight lines 2. and A pair of intersecting straight lines 3. and A point Case II: When In this case equation (i) represents the non-degenerate conic whose nature is given in the following table. S. No. Condition Nature of conic 1. A circle 2. A parabola 3. An ellipse 4. A hyperbola 5. and A rectangular hyperbola 5.0.5. Method to find centre of a Conic. Let be the given conic. Find Solve for x, y we shall get the required centre (x, y) Example: 1 The equation represents (a) A parabola (b) An ellipse (c) A hyperbola (d) A circle Solution: (a) Comparing the given equation with Here, a = 1, b = 1, h = – 1, , f = 0, c = 2 Now i.e., and i.e., So given equation represents a parabola. Example: 2 The centre of is [BIT Ranchi1986] (a) (2, 3) (b) (2, –3) (c) (–2, 3) (d) (–2, –3) Solution: (a) Centre of conic is Here, , , , , Centre Centre .

Comments

Post a Comment

Popular posts from this blog

PHYSICS-15-10- 11th (PQRS) SOLUTION

XI (PQRS) PHYSICS REVIEW TEST-5 Select the correct alternative. (Only one is correct) [15 × 3 = 45] There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. Q.31 Two trains, which are moving along different tracks in opposite directions towards each other, are put on the same track due to a mistake. Their drivers, on noticing the mistake, start slowing down the trains when the trains are 300 m apart. Graphs given below show their velocities as function of time as the trains slow down. The separation between the trains when both have stopped, is: (A) 120 m (B) 280 m (C) 60 m (D*) 20 m 1 [Sol: x1 = 2 1 × 10 × 40 = 200 m; x2 = – 2 × 8 × 20 = –80 m  xreletive = x1 + x2 = 280  separation = 300 – 280 = 20 m (D) ] Q.32 One end of a thin uniform rod of length L and mass M is riveted to the centre of a uniform circular disc of radius r and mass 2 M so that the rod is normal to the disc. The centre of mass of the combination from the centre of the disc is at dist
Read more

8-Circular Motion

8.1 (D) The maximum angular speed of the hoop corresponds to the situation when the bead is just about to slide upwards. The free body diagram of the bead is For the bead not to slide upwards. m2 (r sin 45°) cos 45° – mg sin 45° < N (1) where N = mg cos 45° + m2 (r sin 45ยบ) sin 45° (2) From 1 and 2 we get.  = rad / s. 8.2 (C) Let v be the speed of particle at B, just when it is about to loose contact. From application of Newton's second law to the particle normal to the spherical surface. mv2 = mg sin  (1) r Applying conservation of energy as the block moves from A to B.. 1 mv2 = mg (r cos  – r sin ) (2) 2 Solving 1 and 2 we get 3 sin  = 2 cos  8.3 (A) As the mass is at the verge of slipping ∴ mg sin37 –  mg cos37 = m2r 6 – 8 = 4.5   = 3 16 8.4 (B) As when they collide vt  1  72v 2  2 2  25R   R = vt  t =   5R 6v Now angle covered by A =   vt R 11 Put t  angle covered by A = 6 8.5 (C) The acceleration vector s
Read more

4. THEORY-Current Electrictricity

CURRENT ELECTRICITY  1 . ELECTRIC CURRENT (a) Time rate of flow of charge through a cross sectional area is called Current. if q charge flows in time interval t then average current is given by q Iav = t and Instantaneous current i =. lim q  dq t0 t dt (b) Direction of current is along the direction of flow of positive charge or opposite to the direction of flow of negative charge. But the current is a scalar quantity. i i q+ velocity q– velocity SI unit of current is ampere and 1 Ampere = 1 coloumb/sec 1 coloumb/sec = 1A 2 . CONDUCTOR In some materials, the outer electrons of each atom or molecule are only weakly bound to it. These electrons are almost free to move throughout the body of the material and are called free electrons. They are also known as conduction electrons. When such a material is placed in an electric field, the free electrons drift in a direction opposite to the field. Such materials are called conductors. 3 . INSULATOR Another class o
Read more