Tiêu đề 9. CHAPTER.pdf ✅

9. CONIC SECTIONS (1.) The ellipse 2 2 x y + = 4 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4,0) . Then the equation of the ellipse is [AIEEE-2009] (a.) 2 2 x y + = 12 16 (b.) 2 2 4 48 48 x y + = (c.) 2 2 4 64 48 x y + = (d.) 2 2 x y + = 16 16 (2.) If two tangents drawn from a point P to the parabola 2 y x = 4 are at right angles, then the locus of p is [AIEEE-2010] (a.) x =1 (b.) 2 1 0 x + = (c.) x =−1 (d.) 2 1 0 x − = (3.) The equation of the hyperbola whose foci are (−2,0) and (2,0) and eccentricity is 2 is given by: [AIEEE-2011] (a.) 2 2 − + = x y3 3 (b.) 2 2 − + = 3 3 x y (c.) 2 2 x y − = 3 3 (d.) 2 2 3 3 x y − = (4.) Statement 1 : An equation of a common tangent to the hyperbola 2 y x =16 3 and the ellipse 2 2 2 4 x y + = is y x = + 2 2 3 . [AIEEE-2012] Statement 2: If the line ( ) 4 3 y mx m = +  , 0 m is a common tangent to the parabola 2 y x =16 3 and the ellipse 2 2 2 4 x y + = , then m satisfies 4 2 m m + = 2 24. (a.) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1 (b.) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 (c.) Statement 1 is true,Statement 2 is false (d.) Statement 1 is false, Statement 2 istrue (5.) An ellipse is drawn by taking a diameter of the circle 2 2 ( 1) 1 x y −+= as its semi- minor axis and a diameter of the circle 2 2 x y + − = ( 2) 4 as its semi-major axis. If the centre of the ellipse is at the original and its axes are the coordinate axes, then the equation of the ellipse is [AIEEE-2012] (a.) 2 2 x y + = 4 8 (b.) 2 2 4 8 x y + = (c.) 2 2 x y + = 4 16 (d.) 2 2 4 4 x y + = (6.) Given : A circle, 2 2 2 2 5 x y + = and a parabola, 2 y x = 4 5 Statement-I : An equation of a common tangent to these curves is y x = + 5 . Statement-II : If the line, ( ) 5 y mx m = +  0 m is their common tangent, then m satisfies 4 2 m m −3 + =2 0. [JEE (Main)-2013] (a.) Statement-I is true; statement-II is true; statement-II is a correct explanation for statement-I
(b.) Statement-I is true; statement-II is true; statement-II is not a correct explanation for statement-1 (c.) Statement-I is true; statement-II is false (d.) Statement-I is false; statement-II, is true (7.) The locus of the foot of perpendicular drawn from the centre of the ellipse 2 2 x y + = 3 6 on any tangent to it is [JEE (Main)-2014] (a.) ( ) 2 2 2 2 2 x y x y + = + 6 2 (b.) ( ) 2 2 2 2 2 x y x y + = − 6 2 (c.) ( ) 2 2 2 2 2 x y x y − = + 6 2 (d.) ( ) 2 2 2 2 2 x y x y − = − 6 2 (8.) The slope of the line touching both the parabolas 2 y x = 4 and 2 x y = −32 is [JEE (Main)-2014] (a.) 1 8 (b.) 2 3 (c.) 1 2 (d.) 3 2 (9.) The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse 2 2 1 9 5 + = x y , is [JEE (Main)-2015] (a.) 27 4 (b.) 18 (c.) 27 2 (d.) 27 (10.) Let O be the vertex and Q be any point on the parabola, 2 x y = 8 . If the point P divides the line segment OQ internally in the ratio 1:3 , then the locus of P is [JEE (Main)-2015] (a.) 2 x y = (b.) 2 y x = (c.) 2 y x = 2 (d.) 2 x y = 2 (11.) Let P be the point on the parabola, 2 y x = 8 which is at a minimum distance from the centre C of the circle, 2 2 x y + + = ( 6) 1 . Then the equation of the circle, passing through C and having its centre at P is [JEE (Main)-2016] (a.) 2 2 x y x y + − + − = 4 12 0 (b.) 2 2 2 24 0 4 + − + − = x x y y (c.) 2 2 x y x y + − + + = 4 9 18 0 (d.) 2 2 x y x y + − + + = 4 8 12 0 (12.) The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is [JEE (Main)-2016] (a.) 4 3 (b.) 2 3 (c.) 3 (d.) 4 3
(13.) The eccentricity of an ellipse whose centre is at the origin is 1 2 . If one of its directrices is x =−4 , then the equation of the normal to it at 3 1, 2       is [JEE (Main)- 2017] (a.) 4 2 1 x y − = (b.) 4 2 7 x y + = (c.) x y + = 2 4 (d.) 2 2 y x − = (14.) A hyperbola passes through the point P( 2, 3) and has foci at (2,0) . Then the tangent to this hyperbola at P also passes through the point [JEE (Main)-2017] (a.) (2 2,3 3) (b.) ( 3, 2 ) (c.) (− − 2, 3) (d.) (3 2,2 3) (15.) Two sets A and B are as under : A a b R R a =   −  { , : 5 1 ( ) and b −  5 1} ( )  2 2 B a b R R a b =   − + −  , : 4( 6) 9( 5) 36 , then [JEE (Main)-2018] (a.) B A  (b.) A B  (c.) A B  = (an empty set) (d.) Neither A B  nor B A  (16.) Tangent and normal are drawn at P(16,16) on the parabola 2 y x =16 , which intersect the axis of the parabola at A and B , respectively. If C is the centre of the circle through the points P A, and B and   CPB = , then a value of tan is [JEE (Main)-2018] (a.) 1 2 (b.) 2 (c.) 3 (d.) 4 3 (17.) Tangents are drawn to the hyperbola 2 2 4 36 x y − = at the points P and Q . If these tangents intersect at the point T (0,3) then the area (in sq. units) of PTQ is [JEE (Main)-2018] (a.) 45 5 (b.) 54 3 (c.) 60 3 (d.) 36 5 (18.) Let 0 2     . If the eccentricity of the hyperbola 2 2 2 2 1 cos sin   − = x y is greater than 2, then the length of its latus rectum lies in the interval [JEE (Main)-2019] (a.) (2,3 (b.) (3 / 2, 2 (c.) (1,3 / 2 (d.) (3, ) (19.) Axis of a parabola lies along x -axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x -axis then which of the following points does not lie on it? [JEE (Main)-2019] (a.) (4, 4− ) (b.) (5, 2 6 ) (c.) (6, 4 2 ) (d.) (8,6)
(20.) Let A(4, 4− ) and B(9,6) be points on the parabola, 2 y x = 4 . Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of ACB is maximum. Then, the area (in sq. units) of ACB , is [JEE (Main)-2019] (a.) 32 (b.) 3 31 4 (c.) 1 31 4 (d.) 1 30 2 (21.) A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x -axis. Then the eccentricity of the hyperbola is [JEE (Main)-2019] (a.) 3 2 (b.) 3 (c.) 2 3 (d.) 2 (22.) If the parabolas ( ) 2 y b x c = − 4 and 2 y ax = 8 have a common normal, then which one of the following is a valid choice for the ordered triad (abc , , ) ? [JEE (Main)- 2019] (a.) 1 , 2,0 2       (b.) 1 , 2,3 2       (c.) (1,1,0) (d.) (1,1,3) (23.) The equation of a tangent to the hyperbola 2 2 4 5 20 x y − = parallel to the line x y − = 2 is [JEE (Main)-2019] (a.) x y −+= 7 0 (b.) x y − + =1 0 (c.) x y − − =3 0 (d.) x y − + =9 0 (24.) The length of the chord of the parabola 2 x y = 4 having equation x y − + = 2 4 2 0 is [JEE (Main)-2019] (a.) 3 2 (b.) 6 3 (c.) 2 11 (d.) 8 2 (25.) Let ( ) 2 2 2 , : 1 1 1   =  − =     + − y x S x y R r r , where r 1 . Then S represents [JEE (Main)-2019] (a.) An ellipse whose eccentricity is 1 , when 1. 1  + r r (b.) An ellipse whose eccentricity is 2 , when 1. 1  + r r (c.) A hyperbola whose eccentricity is 2 , 0 1 1 when r r   + (d.) A hyperbola whose eccentricity is