ANALYTICAL GEOMETRY ONE MARK QUESTIONS PREPARED BY: R.RAJENDRAN. M.A., M. Sc., M. Ed., K.C.SANKARALINGA NADAR HR. SEC. SCHOOL, CHENNAI-21

CHOOSE THE CORRECT ANSWER The axis of the parabola y2 – 2y + 8x – 23 = 0 is (a) y = – 1 (b) x = – 3 (c) x = 3 (d) y = 1 16x2 – 3y2 – 32x – 12y – 44 = 0 represents (a) an ellipse (b) a circle (c) a parabola (d) a hyperbola

Choose the correct answer The line 4x + 2y = c is a tangent to the parabola y2 = 16x then c is (a) – 1 (b) – 2 (c) 4 (d) – 4 The point of intersection of the tangent at t1 = t and t2 = 3t to the parabola y2 = 8x is (a) (6t2, 8t) (b) (8t, 6t2) (c) (t2, 4t) (d) (4t, t2)

Choose the correct answer The length of the latus rectum of the parabola y2 – 4x + 4y + 8 = 0 is (a) 8 (b) 6 (c) 4 (d) 2 The directrix of the parabola y2 = x + 4 is (a) x = 15/4 (b) x = – 15/4 (c) x = – 17/4 (d) x = 17/4

Choose the correct answer The length of the latus rectum of the parabola whose vertex is (2, –3) and the directrix x = 4 is (a) 2 (b) 4 (c) 6 (d) 8 The focus of the parabola x2 = 16y is (a) (4, 0) (b) (0, 4) (c) (– 4, 0) (d) (0, – 4)

Choose the correct answer The tangent at the end of any focal chord to the parabola y2 = 12x intersect on the line (a) x – 3 = 0 (b) x + 3 = 0 (c) y + 3 = 0 (d) y – 3 = 0 The vertex of the parabola x2 = 8y – 1 is (a) (b) (c) (d)

Choose the correct answer The angle between the two tangents drawn from the point (– 4, 4) to the parabola y2 = 16x is (a) 45° (b) 30° (c) 60° (d) 90° The line 2x + 6y + 9 = 0 touches the parabola y2 = 8x at the point (a) (0, – 3) (b) (2, 4) (c) (d)

Choose the correct answer The eccentricity of the conic 9x2 + 5y2 – 54x – 40y + 116 = 0 is (a) 1/3 (b) 2/3 (c) 4/9 (d) 2/5 The length of the semi-major and the length of minor-axis of the ellipse (a) 26, 12 (b) 13, 24 (c) 12, 26 (d) 13, 12

Choose the correct answer The distance between the foci of the ellipse 9x2 + 5y2 = 180 is (a) 4 (b) 6 (c) 8 (d) 2 If the length of the major and semi minor-axes of an ellipse are 8, 2 and their corresponding equations are y – 6 = 0 and x + 4 = 0 then the equations of the ellipse is (a) (b) (c) (d)

Choose the correct answer The straight line 2x – y + c = 0 is a tangent to the ellipse 4x2 + 8y2 = 32 if c is (a) 23 (b) 6 (c) 36 (d) 4 The sum of the distance of any point on the ellipse 4x2 + 9y2 = 36 from (5, 0)( –5, 0)is (a) 4 (b) 8 (c) 6 (d) 18

Choose the correct answer If the normal at the end of the latus rectum of the ellipse x2 + 3y2 = 12 intersect the major axis at G, then CG is (a) 16/3 (b) 82/3 (c) 42/3 (d) 32/3 The radius of the director circle of the conic 9x2 + 16y2 = 144 is (a) 7 (b) 4 (c) 3 (d) 5

Choose the correct answer The locus of foot of the perpendicular from the focus to a tangent of the curve 16x2 + 25y2 = 400 is (a) x2 + y2 = 4 (b) x2 + y2 = 25 (c) x2 + y2 = 16 (d) x2 + y2 = 9 The eccentricity of the hyperbola 12x2 – 4y2 – 24x + 48y – 127 = 0 is (a) 4 (b) 3 (c) 2 (d) 6

Choose the correct answer The tangent at any point P on the ellipse whose center C meets the major axis at T and PN is the perpendicular to the major axis then CN.CT = (a) 6 (b) 3 (c) 3 (d) 6 If Band B’ are the ends of the major axis, F1 and F2 are the foci of the ellipse then the area of F1BF2B’ is (a) 16 (b) 8 (c) 162 (d) 322

Choose the correct answer The eccentricity of the hyperbola whose latus rectum is equal to half of its conjugate axis is (a) 3/2 (b) 5/3 (c) 5/2 (d) 3/2 The directrix of the hyperbola x2 – 4(y – 3)2 = 16 is (a) (b) (c) (d)

Choose the correct answer The difference between the focal distances of any point on the hyperbola is 24 and the eccentricity is 2. Then the equation of the hyperbola is (a) (b) (c) (d)

Choose the correct answer The line 5x – 2y + 4k = 0 is tangent to the hyperbola 4x2 – y2 = 36 then k is (a) 4/9 (b) 2/3 (c) 9/4 (d) 81/16 The equation of the chord of contact of tangents from (2, 1) to the hyperbola is (a) 9x – 8y – 72 = 0 (b) 9x + 8y + 72 = 0 (c) 8x – 9y – 72 = 0 (d) 8x + 9y + 72 = 0

Choose the correct answer The angle between the asymptotes to the hyperbola is (a) (b) (c) (d) Length of the semi-transverse axis of the rectangular hyperbola xy = 8 is (a) 2 (b) 4 (c) 16 (d) 8

Choose the correct answer The asymptotes of the hyperbola 36y2 – 25x2 + 900 = 0 is (a) (b) (c) (d) The asymptotes of the rectangular hyperbola xy = c2 are (a) x = c, y = c (b) x = 0, y = c (c) x = c, y = 0 (d) x = 0, y = 0

Choose the correct answer The product of the perpendicular drawn from the point (8, 0) on the hyperbola to its asymptotes is (a) (b) (c) (d) The area of the triangle formed by the tangent at any point on the rectangular hyperbola xy = 72 and its asymptotes is (a) 36 (b) 18 (c) 72 (d) 144

Choose the correct answer P is any point on the hyperbola the ordinate at P meets the asymptotes in Q and Q’ then QP.Q’P is (a) 36 (b) 6 (c) 4 (d) 2 The locus of the point of intersection of perpendicular tangents to the hyperbola is (a) x2 + y2 = 25 (b) x2 + y2 = 4 (c) x2 + y2 = 3 (d) x2 + y2 = 7

Choose the correct answer The eccentricity of the hyperbola with asymptotes x + 2y – 5 = 0 and 2x – y + 5 = 0 is (a) 3 (b) 2 (c) 3 (d) 2 The co-ordinate of the vertices of the rectangular hyperbola xy = 16 are (a) (4, 4), (– 4, –4) (b) (2, 8), (–2. –8) (c) (4, 0), (–4, 0) (d) (8, 0), (–8, 0)

Choose the correct answer The length of the latus rectum of the rectangular hyperbola xy = 32 is (a) 82 (b) 32 (c) 8 (d) 16 The normal to the rectangular hyperbola xy = 9 at (6, 3/2) meets the curve again at (a) (3/8, 4) (b) (–24, –3/8) (c) (–3/8, –24) (d) (24, 3/8)

Choose the correct answer The axis of the parabola y2 = 4x is (a) x = 0 (b) y = 0 (c) x = 1 (d) y = 1 The vertex of the parabola y2 = 4x is (a) (1, 0) (b) (0, 1) (c) (0, 0) (d) (0, – 1) The focus of the parabola y2 = 4x is (a) (0, 1) (b) (1, 1) (c) (0, 0) (d) (1, 0)

Choose the correct answer The directrix of the parabola y2 = 4x is (a) y = – 1 (b) x = – 1 (c) y = 1 (d) x = 1 The equation of the latus rectum of the parabola y2 = 4x is (a) x = 1 (b) y = 1 (c) x = 4 (d) y = – 1 The length of the LR of y2 = 4x is (a) 2 (b) 3 (c) 1 (d) 4

Choose the correct answer The axis of the parabola x2 = – 4y is (a) y = 1 (b) x = 0 (c) y = 0 (d) x = 1 The vertex of the parabola x2 = – 4y is (a) (0, 1) (b) (0, –1) (c) (1, 0) (d) (0, 0) The focus of the parabola x2 = – 4y is (a) (0, 0) (b) (0, –1) (c) (0, 1) (d) (1, 0)

Choose the correct answer The directrix of the parabola x2 = – 4y is (a) x = 1 (b) x = 0 (c) y = 1 (d) y = 0 The equation of the latus rectum of the parabola x2 = – 4y is (a) x = –1 (b) y = –1 (c) x = 1 (d) y = 1 The focus of the parabola y2 = – 8x is (a) (0, – 2) (b) (0, 2) (c) (– 2, 0) (d) (2, 0)

Choose the correct answer The length of the LR of x2 = – 4y is (a) 1 (b) 2 (c) 3 (d) 4 The axis of the parabola y2 = – 8x is (a) x = 0 (b) x = 2 (c) y = 2 (d) y = 0 The vertex of the parabola y2 = – 8x is (a) (0, 0) (b) (2, 0) (c) (0, – 2) (d) (2, – 2)

Choose the correct answer The focus of the parabola y2 = – 8x is (a) (0, – 2) (b) (0, 2) (c) (– 2, 0) (d) (2, 0) The directrix of the parabola y2 = – 8x is (a) y + 2 = 0 (b) x – 2 = 0 (c) y – 2 = 0 (d) x + 2 = 0 The equation of the latus rectum of the parabola y2 = – 8x is (a) y –2 = 0 (b) y + 2 = 0 (c) x – 2 = 0 (d) x + 2 = 0

Choose the correct answer The length of the LR of y2 = – 8x is (a) 8 (b) 6 (c) 4 (d) – 8 The axis of the parabola x2 = 20y is (a) y = 5 (b) x = 5 (c) x = 0 (d) y = 0 The vertex of the parabola x2 = 20y is (a) (0, 5) (b) (0, 0) (c) (5, 0) (d) (0, – 5)

Choose the correct answer The focus of the parabola x2 = 20y is (a) (0, 0) (b) (5, 0) (c) (0, 5) (d) (– 5, 0) The equation of the directrix of the parabola x2 = 20y is (a) y – 5 = 0 (b) x + 5 = 0 (c) x – 5 = 0 (d) y + 5 = 0 The equation of the latus rectum of the parabola x2 = 20y is (a) x – 5 = 0 (b) y – 5 = 0 (c) y + 5 = 0 (d) x + 5 = 0

Choose the correct answer The length of the latus rectum of the parabola x2 = 20y is (a) 20 (b) 10 (c) 5 (d) 4 If the center of the ellipse is (2, 3) one of the foci is (3, 3) then the other focus is (a) (1, 3) (b) (– 1, 3) (c) (1, – 3) (d) (–1, –3) The equation of the LR s of 25x2 + 9y2 = 225 are (a) y =  5 (b) x =  5 (c) y =  4 (d) x =  4

Choose the correct answer The equation of the major and minor axes of are (a) x = 3, y = 2 (b) x = – 3, y = – 2 (c) x = 0, y = 0 (d) y = 0, x = 0 The length of minor and major axes of are (a) 6, 4 (b) 3, 2 (c) 4, 6 (d) 2, 3

Choose the correct answer The equations of the major and minor axes of 4x2 + 3y2 = 12 are (a) x = 3, y = 2 (b) x = – 3, y = – 2 (c) x = 0, y = 0 (d) y = 0, x = 0 The lengths of minor and major axes of the ellipse 4x2 + 3y2 = 12 are (a) 4, 23 (b) 2, 3 (c) 23, 4 (d) 2, 3

Choose the correct answer The equation of the directrices of are (a) (b) (c) (d) The equation of the directrices of 25x2 + 9y2 = 225 are (a) (b) (c) (d)

Choose the correct answer The equation of the latus rectum of are (a) y = 7 (b) x = 7 (c) x = 7 (d) y = 7 If the center of the ellipse is (4, – 2) and one of the foci is (4, 2), then the other focus is (a) (4, 6) (b) (6, – 4) (c) (4, – 6) (d) (6, 4)

Choose the correct answer The length of the LR of is (a) (b) (c) (d) The length of the LR of 25x2 + 9y2 = 225 is (a) (b) (c) (d)

Choose the correct answer The eccentricity of the ellipse is (a) 1/5 (b) 3/5 (c) 2/5 (d) 4/5 The eccentricity of the ellipse is (a) 5/3 (b) 3/5 (c) 3/5 (d) 2/3

Choose the correct answer The eccentricity of the ellipse 16x2 + 25y2 = 400 is (a) 4/5 (b) 3/5 (c) ¾ (d) 2/5 Center of the ellipse is (a) (0, 0) (b) (5, 0) (c) (3, 5) (d) (0, 5)

Choose the correct answer The center of the ellipse is (a) (0, 3) (b) (2, 3) (c) (0, 0) (d) (3, 0) The foci of the ellipse are (a) (0,  5) (b) (0,  4) (c) ( 5, 0) (d) ( 4, 0)

Choose the correct answer The foci of the ellipse are (a) ( 5, 0) (b) (0,  5) (c) (0,  5) (d) ( 5, 0) The foci of the ellipse 16x2 + 25y2 = 400 are (a) ( 3, 0) (b) (0,  3) (c) (0,  5) (d) ( 5, 0)

Choose the correct answer The vertices of the ellipse are (a) (0,  5) (b) (0,  3) (c) ( 5, 0) (d) ( 3, 0) The vertices of the ellipse are (a) (0,  3) (b) ( 2, 0) (c) ( 3, 0) (d) (0,  2)

Choose the correct answer The vertices of the ellipse 16x2 + 25y2 = 400 are (a) (0,  4) (b) ( 5, 0) (c) ( 4, 0) (d) (0,  5) The equations of transverse and conjugate axes of the hyperbola are (a) x = 2, y = 3 (b) y = 0, x = 0 (c) x = 3, y = 2 (d) x = 0, y = 0

Choose the correct answer The equations of transverse and conjugate axes of the hyperbola 16x2 – 9y2 = 144 are (a) y = 0, x = 0 (b) x = 3, y = 4 (c) x = 0, y = 0 (d) y = 3, x = 4 The equations of transverse and conjugate axes of the hyperbola 144x2 – 25y2 = 3600 are (a) y = 0, x = 0 (b) x = 12, y = 5 (c) x = 0, y = 0 (d) x = 5, y = 12

Choose the correct answer The equations of transverse and conjugate axes of the hyperbola 8y2 – 2x2 = 16 are (a) x = 22, y = 2 (b) x = 2, y = 22 (c) x = 0, y = 0 (d) y = 0, x = 0 The equations of the directrices of the hyperbola are (a) y = (b) x = (c) y = (d) x =

Choose the correct answer The equations of the directrices of the hyperbola 16x2 – 9y2 = 144 are (a) x =  (b) y =  (c) x =  (d) y =  The equations of the latus rectum of the hyperbola are (a) y =  13 (b) y = 13 (c) x =  13 (d) x =  13

Choose the correct answer The equations of the LR of the hyperbola 16x2 – 9y2 = 144 are (a) y =  5 (b) x =  5 (c) y =  5 (d) x =  5 The length of the LR of the hyperbola are (a) 4/3 (b) 8/3 (c) 3/2 (d) 9/4

Choose the correct answer The eccentricity of the hyperbola are (a) 34/3 (b) 5/3 (c) 34/3 (d) 34/5 The eccentricity of the hyperbola 25x2 – 16y2 = 400 is (a) (0, 4) (b) (0, 5) (c) (4, 5) (d) (0, 0)

Choose the correct answer The foci of the hyperbola are (a) (0, 34) (b) ( 34, 0) (c) (0,  34) (d) ( 34, 0) The vertices of the hyperbola 25x2 – 16y2 = 400 are (a) (0,  4) (b) ( 4, 0) (c) (0,  5) (d) ( 5, 0)

Choose the correct answer The equation of the tangent at (3, – 6) to the parabola y2 = 12x is (a) x – y – 3 = 0 (b) x + y – 3 = 0 (c) x – y + 3 = 0 (d) x + y + 3 = 0 The equation of the tangent at (– 3, 1) to the parabola x2 = 9y is (a) 3x – 2y – 3 = 0 (b) 2x – 3y + 3 = 0 (c) 2x + 3y + 3 = 0 (d) 3x + 2y + 3 = 0

Choose the correct answer The equation of chord of contact of tangents from the point (– 3, 1) to the parabola y2 = 8x is (a) 4x – y – 12 = 0 (b) 4x + y + 12 = 0 (c) 4y – x – 12 = 0 (d) 4y – x + 12 = 0 The equation of chord of contact of tangents from (2, 4) to the ellipse 2x2 + 5y2 = 20 is (a) x – 5y + 5 = 0 (b) 5x – y + 5 = 0 (c) x + 5y – 5 = 0 (d) 5x – y – 5 = 0

Choose the correct answer The equation of chord of contact of tangents from (5, 3) to the hyperbola 4x2 – 6y2 = 24 is (a) 9x + 10y + 12 = 0 (b) 10x + 9y – 12 = 0 (c) 9x – 10y + 12 = 0 (d) 10x – 9y – 12 = 0 The combined equation of the asymptotes to the hyperbola 36x2 – 25y2 = 900 is (a) 25x2 + 36y2 = 0 (b) 36x2 – 25y2 = 0 (c) 36x2 + 25y2 = 0 (d) 25x2 – 36y2 = 0

Choose the correct answer The angle between the asymptotes of the hyperbola 24x2 – 8y2 = 27 is (a) /3 (b) /3 or 2/3 (c) 2/3 (d) – 2/3 The point of contact of the tangent y = mx + c and the parabola y2 = 4ax is (a) (b) (c) (d)

Choose the correct answer The point of contact of the tangent y = mx + c and the ellipse is (a) (b) (c) (d) If t1 and t2 are the extremities of any focal chord of a parabola y2 = 4ax then t1t2 is (a) – 1 (b) 0 (c) 1 (d) ½

Choose the correct answer The true statements of the following are (1) Two tangents and 3 normals can be drawn to a parabola from a point (2) Two tangents and 4 normals can be drawn to an ellipse from a point (3) Two tangents and 4 normals can be drawn to a hyperbola from a point (4) Two tangents and 4 normals can be drawn to a RH from a point (a) (1), (2), (3) and (4) (b) (1) and (2) only (c) (3) and (4) only (d) (1), (2) and (3)

Choose the correct answer The normal at t1 on the parabola y2 = 4ax meets the parabola at t2 then is (a) – t2 (b) t2 (c) t1 + t2 (d) 1/t2

Choose the correct answer The condition that the line lx + my + n = 0 may be a normal to the ellipse is (a) at3 + 2alm2 + m2n = 0 (b) (c) (d)

Choose the correct answer The condition that the line lx + my + n = 0 may be a normal to the hyperbola is (a) at3 + 2alm2 + m2n = 0 (b) (c) (d)

Choose the correct answer The condition that the line lx + my + n = 0 may be a normal to the parabola y2 = 4ax is (a) at3 + 2alm2 + m2n = 0 (b) (c) (d)

Choose the correct answer The chord of contact of tangents from any point on the directrix of the parabola y2 = 4ax passes though its ------------ (a) vertex (b) focus (c) directrix (d) latus rectum The chord of contact of tangents from any point on the directrix of the ellipse passes though its – (a) vertex (b) focus (c) directrix (d) latus rectum

Choose the correct answer The chord of contact of tangents from any point on the directrix of the hyperbola passes though its ------------ (a) vertex (b) focus (c) directrix (d) latus rectum The point of intersection of tangents at t1 and t2 to the parabola y2 = 4ax is (a) (a(t1 + t2), at1t2) (b) (at1t2, a(t1 + t2)) (c) (at2, 2at) (d) (at1t2, a(t1 – t 2)

Choose the correct answer The normal at the point t1 on the parabola y2 = 4ax meets the parabola again at t2 then – t2 is (a) t12 (b) 1/t1 (c) t1 (d) t1 + 2/t1 The locus of the foot of perpendicular from the focus on any tangents to the hyperbola is (a) x2 + y2 = a2 – b2 (b) x2 + y2 = a2 (c) x2 + y2 = a2 + b2 (d) x2 + y2 = 0

Choose the correct answer The locus of the point of intersection of perpendicular tangents to the parabola y2 = 4ax is (a) latus rectum (b) directrix (c) tangent at the vertex (d) axis of the parabola The locus of the foot of perpendicular from the focus on any tangents to the ellipse is (a) x2 + y2 = a2 – b2 (b) x2 + y2 = a2 (c) x2 + y2 = a2 + b2 (d) x2 + y2 = 0

Choose the correct answer The locus of the foot of perpendicular from the focus on any tangents to the parabola y2 = 4ax is (a) x2 + y2 = a2 – b2 (b) x2 + y2 = a2 (c) x2 + y2 = a2 + b2 (d) x = 0 The locus of point of intersection of perpendicular tangents to the ellipse is (a) x2 + y2 = a2 – b2 (b) x2 + y2 = a2 (c) x2 + y2 = a2 + b2 (d) x2 + y2 = 0

Choose the correct answer The locus of point of intersection of perpendicular tangents to the hyperbola is (a) x2 + y2 = a2 – b2 (b) x2 + y2 = a2 (c) x2 + y2 = a2 + b2 (d) x2 + y2 = 0 The condition that the line lx + my + n = 0 may be a tangent to the parabola y2 = 4ax is (a) a2 l2 + b2 m2 = n2 (b) am2 = ln (c) a2 l2 – b2 m2 = n2 (d) 4c2 lm = n2

Choose the correct answer The condition that the line lx + my + n = 0 may be a tangent to the ellipse is (a) a2 l2 + b2 m2 = n2 (b) am2 = ln (c) a2 l2 – b2 m2 = n2 (d) 4c2 lm = n2 The condition that the line lx + my + n = 0 may be a tangent to the hyperbola is (a) a2 l2 + b2 m2 = n2 (b) am2 = ln (c) a2 l2 – b2 m2 = n2 (d) 4c2 lm = n2

Choose the correct answer The condition that the line lx + my + n = 0 may be a tangent to the rectangular hyperbola xy = c2 is (a) a2 l2 + b2 m2 = n2 (b) am2 = ln (c) a2 l2 – b2 m2 = n2 (d) 4c2 lm = n2 If t1 and t2 are extremities of any focal chord of the parabola y2 = 4ax then t1t2 is (a) 1 (b) – 1 (c) 0 (d) – 2

Choose the correct answer The foot of a perpendicular from a focus of the hyperbola on an asymptote lies on the ------- (a) centre (b) corresponding directrix (c) vertex (d) latus rectum If the normal to the RH xy = c2 at t1 meets the curve again at t2 then t13 t2 = (a) 1 (b) 0 (c) – 1 (d) – 2

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