SELECT THE CORRECT ALTERNATIVE (ONLY ONE CORRECT

EXERCISE - 03MATCH THE COLUMNFollowing questions contains statements given in two columns, which have to be matched. The statements inColumn-I are labelled as A, B, C and D while the statements in Column-II are labelled as p, q, r and s. Any givenstatement in Column-I can have correct matching with ONE statement in Column-II.1.Column-I(A)(B)Column-IIThe normal chord at a point t on the parabola y 2 4xsubtends a right angle at the vertex, then t 2 isThe area of the triangle inscribed in the curve y 2 4x.If the parameter of vertices are 1, 2 and 4 is(p)4(q)2(C)11 1The number of distinct normal possible from , to the 4 4 (r)3(D)parabola y 2 4x isThe normal at (a, 2a) on y 2 4ax meets the curve againat (at 2, 2at), then the value of t – 1 is(s)62.Column-IColumn-II(A)Area of a triangle formed by the tangents drawn from apoint (–2, 2) to the parabola y 2 4(x y) and theircorresponding chord of contact is(p)8(B)Length of the latus rectum of the conic(q)4 3(r)4(s)24/52(C)(D)2225{(x – 2) (y – 3) } (3x 4y – 6) isIf focal distance of a point on the parabola y x 2 – 4 is 25/4and points are of the form ( a , b) then value of a b isLength of side of an equilateral triangle inscribedin a parabola y 2 – 2x – 2y – 3 0 whose oneangular point is vertex of the parabola, isASSERTION&REASONThese questions contain, Statement-I (assertion) and Statement-II ment-IStatement-Iisisisistrue, Statement-II is true ; Statement-II is correct explanation for Statement-I.true, Statement-II is true ; Statement-II is NOT a correct explanation for Statement-I.true, Statement-II is false.false, Statement-II is true.Statement-I : If normal at the ends of double ordinate x 4 of parabola y 2 4x meet the curve againat P and P' respectively, then PP' 12 unit.BecauseStatement-II : If normal at t 1 of y 2 4ax meets the parabola again at t 2 , then t 1 2 2 t 1 t 2 .(A) A(B) B(C) C(D) D2.Statement-I : The lines from the vertex to the two extremities of a focal chord of the parabola y 2 4axare at an angle of .2BecauseStatement-II : If extremities of focal chord of parabola are ( at12 , 2at 1 ) and ( at 22 , 2at 2 ), then t 1 t 2 –1(A) A(B) B(C) C(D) DPage 6 of 14

3.Statement-I : If P 1 Q 1 and P 2 Q 2 are two focal chords of the parabola y 2 4ax, then the locus of pointof intersection of chords P1P2 and Q1Q2 is directrix of the parabola. Here P1P2 and Q1Q2 are not parallel.BecauseStatement-II : The locus of point of intersection of perpendicular tangents of parabola is directrix of parabola.(A) A(B) B(C) C(D) DCOMPREHENSION BASED QUESTIONSComprehension # 1Observe the following facts for a parabola :(i)Axis of the parabola is the only line which can be the perpendicular bisector of the two chords of theparabola.(ii)If AB and CD are two parallel chords of the parabola and the normals at A and B intersect at P and thenormals at C and D intersect at Q, then PQ is a normal to the parabola.Let a parabola is passing through (0, 1), (–1, 3), (3, 3) & (2, 1)On t he basis of above i nformat ion, a nswer t he fol low i ng que st ions :1.The vertex of the parabola is -2. 1 1 (A) 1, (B) , 1 3 3 The directrix of the parabola is -(C) (1, 3)(D) (3, 1)1111 0 0(B) y 0(C) y 0(D) y 241224223.For the parabola y 4x, AB and CD are any two parallel chords having slope 1. C1 is a circle passing throughO, A and B and C2 is a circle passing through O, C and D, where O is origin. C1 and C2 intersect at (A) (4, –4)(B) (–4, 4)(C) (4, 4)(D) (–4, –4)Comprehension # 2 :If a source of light is placed at the fixed point of a parabola and if the parabola is a reflecting surface, thenthe ray will bounce back in a line parallel to the axis of the parabola.On t he basis of above i nformat ion, a nswer t he fol low i ng que st ions :1.A ray of light is coming along the line y 2 from the positive direction of x-axis and strikes a concave mirrorwhose intersection with the xy-plane is a parabola y 2 8x, then the equation of the reflected ray is (A) 2x 5y 4(B) 3x 2y 6(C) 4x 3y 8(D) 5x 4y 102.A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation isy 2 10y – 4x 17 0 After reflection, the ray must pass through the point (A) (–2, –5)(B) (–1, –5)(C) (–3, –5)(D) (–4, –5)3.Two ray of light coming along the lines y 1 and y –2 from the positive direction of x-axis and strikesa concave mirror whose intersection with the xy-plane is a parabola y 2 x at A and B respectively. Thereflected rays pass through a fixed point C, then the area of the triangle ABC is (A) y –(A)21sq. unit8(B)19sq. unit2ANSWERM ISCEL L AN E OU S TYP E Q U ESTION (C)15sq. unit2E XE R CISE -32. (A) (r); (B) (s); (C) (p); (D) (q)A s s er ti o n & R eas o n1. C KEY(D)Matc h th e C o lu mn1. (A) (q); (B) (s); (C) (q); (D) (p) 17sq. unit22. DC o mp rehe ns i o nB as ed3. BQu e st i o nsC o mp re he n s i o n # 1 :1. A2. C3. AC o mp re he n s i o n # 2 :1. C2. B3. APage 7 of 14

EXERCISE - 04 [A]1.Find the equation of parabola, whose focus is (–3, 0) and directrix is x 5 0.2.Find the vertex, axis, focus, directrix, latus rectum of the parabola x2 2y – 3x 5 03.Find the equation of the parabola whose focus is (1, –1) and whose vertex is (2, 1). Also find its axis and latusrectum.4.If the end points P(t1) and Q( t2) of a chord of a parabola y2 4ax satisfy the relation t1t2 k (constant) thenprove that the chord always passes through a fixed point. Find that point also ?5.Find the locus of the middle points of all chords of the parabola y2 4ax which are drawn through the vertex.6.O is the vertex of the parabola y² 4ax & L is the upper end of the latus rectum. If LH is drawn perpendicularto OL meeting OX in H , prove that the length of the double ordinate through H is 4a 5 .7.Find the length of the side of an equilateral triangle inscribed in the parabola, y2 4x so that one of its angularpoint is at the vertex.8.Two perpendicular chords are drawn from the origin 'O' to the parabola y x2, which meet the parabola at Pand Q. Rectangle POQR is completed. Find the locus of vertex R.9.Find the set of values of in the interval [ /2, 3 /2], for which the point (sin , cos ) does not lie outside theparabola 2y2 x – 2 0.10.Find the length of the focal chord of the parabola y2 4ax whose distance from the vertex is p.11.If 'm' varies then find the range of c for which the line y mx c touches the parabola y2 8(x 2).12.Find the equations of the tangents to the parabola y² 16x, which are parallel & perpendicular respectivelyto the line 2x y 5 0 . Find also the coordinates of their points of contact .13.Find the equations of the tangents of the parabola y² 12x, which passes through the point (2, 5).14.Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabolay2 4ax is y²(2x a) a(3x a)².15.Two tangents to the parabola y² 8x meet the tangent at its vertex in the points P & Q. If PQ 4 units,prove that the locus of the point of the intersection of the two tangents is y² 8 (x 2).16.Find the equation of the circle which passes through the focus of the parabola x2 4y & touches itat the point (6 , 9).17.In the parabola y² 4ax, the tangent at the point P, whose abscissa is equal to the latus rectum meets theaxis in T & the normal at P cuts the parabola again in Q. Prove that PT : PQ 4 : 5.18.Show that the normals at the points (4a , 4a) & at the upper end of the latus rectum of the parabola y² 4axintersect on the same parabola.19.Show that the locus of a poi nt, such that t wo of the three normals draw n from it to the parabolay² 4ax are perpendicular is y² a(x 3a).20.If the normal at P(18, 12) to the parabola y² 8x cuts it again at Q, then show that 9PQ 80 1021.Prove that the locus of the middle point of portion of a normal to y² 4ax intercepted between the curve& the axis is another parabola. Find the vertex & the latus rectum of the second parabola.Page 8 of 14

22.A variable chord PQ of the parabola y² 4x is drawn parallel to the line y x. If the parameters of thepoints P & Q on the parabola are p & q respectively, show that p q 2. Also show that the locus of thepoint of intersection of the normals at P & Q is 2x y 12.23.P & Q are the points of contact of the tangents drawn from the point T to the parabola y² 4ax. IfPQ be the normal to the parabola at P, prove that TP is bisected by the directrix.24.The normal at a point P to the parabola y² 4ax meets its axis at G. Q is another point on the parabolasuchthat QG is perpendicular to the axis of the parabola. Prove that QG² PG² constant.25.Three normals to y² 4x pass through the point (15, 12). Show that one of the normals is given byy x 3 & find the equations of the others.ANSWERCON CEP TUAL SU BJ ECTIVE E X ER CISEKEYEXERCISE-4(A)1.y2 4(x 4)2. 3 11 3 15 Vertex ,, focus ,, 2 8 2 8 3.(2x – y – 3)2 –20(x 2y – 4), axis : 2x – y – 3 0. latus rectum 4 5 .4.(–ak, 0)9. [ / 2, 5 / 6] [ , 3 / 2]12.16.2x y 2 0, (1, 4) ; x 2y 16 0, (16, 16)x2 y2 18 x 28 y 27 02 1 . (a , 0) ; a5.axis : x y2 2ax7.10.37, directrix : y – , latus rectum 2288.8 34a 3p2Page 9 of 14x2 y – 211.(– , –4] [4, )13.3x 2y 4 0 ; x y 3 02 5 . y 4x 72 , y 3x 33

EXERCISE - 04 [B]1.If from the vertex of a parabola a pair of chords be drawn at right angles to one another, & with thesechords as adjacent sides a rectangle be constructed , then find the locus of the outer corner of therectangle.2.Two perpendicular straight lines through the focus of the parabola y² 4ax meet its directrix in T & T respectively. Show that the tangents to the parabola parallel to the perpendicular lines intersect in the midpoint of T T '.3.Find the condition on ‘a’ & ‘b’ so that the two tangents drawn to the parabola y² 4ax from a pointare normals to the parabola x² 4by.4.TP & TQ are tangents to the parabola and the normals at P & Q meet at a point R on the curve. Prove thatthe centre of the circle circumscribing the triangle TPQ lies on the parabola 2 y² a(x a).5.Let S is the focus of the parabola y2 4ax and X the foot of the directrix, PP' is a double ordinate of thecurve and PX meets the curve again in Q. Prove that P'Q passes through focus.6.Prove that on the axis of any parabola y² 4ax there is a certain point K which has the property that , if a chordPQ of the parabola be drawn through it , then1 P K 2 1 Q K 2is same for all positions of the chord. Findalso the coordinates of the point K.7.If (x 1, y 1), (x 2, y 2) and (x 3, y 3) be three points on the parabola y 2 4ax and the normals at these pointsmeet in a point, then prove thatx x3x xx1 x 2 2 3 1 0y3y1y28.A variable chord joining points P(t1) and Q(t2) of the parabola y² 4ax subtends a right angle at a fixed pointt0 of the curve. Show that it passes through a fixed point. Also find the co-ordinates of the fixed point.9.Show that a circle circumscribing the triangle formed by three co-normal points passes through the vertex ofthe parabola and its equation is, 2(x 2 y 2) – 2(h 2a) x – ky 0, where (h, k) is the point from where threeconcurrent normals are drawn.10.A ray of light is coming along the line y b from the positive direction of x-axis & strikes a concavemirror whose intersection with the xy-plane is a parabola y2 4 ax. Find the equation of the reflectedray & show that it passes through the focus of the parabola. Both a & b are positive.BRAIN STOR MIN G SUBJ ECTIVE E X ER CISE1.y² 4a(x 8a)3.ANSWERa² 8b² 6. (2a , 0) 8 .KEY[REE 95]EXERCISE-4(B)[a(t²0 4), 2at0] 10. 4abx (4a² – b²)y – 4a²b 0Page 10 of 14

EXERCISE - 05 [A]1.The length of the latus rectum of the parabola x2 – 4x – 8y 12 0 is(1) 42.(2) 6(3) 8(4) 10The equation of tangents to the parabola y2 4ax at the ends of its latus rectum is(1) x – y a 03.[AIEEE-2002](2) x y a 0(3) x y – a 0[AIEEE-2002](4) both (1) and (2)21The normal at the point (bt , 2bt1) on a parabola meets the parabola again in the point (bt22, 2bt2), then[AIEEE-2003](1) t2 t1 4.2t1(2) t2 –t1 –2t1(3) t2 –t1 2t1(4) t2 t1 –2t1If a 0 and the line 2bx 3cy 4d 0 passes through the points of intersection of the parabolas y2 4ax andx2 4ay, then-[AIEEE-2004](1) d2 (2b 3c)2 05.34(2) xy 3516(4) d2 (3b – 2c)2 0a3 x2a2 x – 2a is23(3) xy [AIEEE-2006]64105(4) xy 10564The equation of a tangent to the parabola y2 8x is y x 2. The point on this line from which the othertangent to the parabola is perpendicular to the given tangents is[AIEEE-2007](1) (–1, 1)7.(3) d2 (2b – 3c)2 0The locus of the vertices of the family of parabolas y (1) xy 6.(2) d2 (3b 2c)2 0(2) (0, 2)(3) (2, 4)(4) (–2, 0)A parabola has the origin as its focus and the line x 2 as the directrix. Then the vertex of the parabola is at [AIEEE-2008](1) (0, 2)8.(2) (1, 0)(3) (0, 1)(4) (2, 0)2If two tangents drawn from a point P to the parabola y 4x are at right angles then the locus of P is :[AIEEE-2010](1) x 19.(2) 2x 1 0(3) x –1(4) 2x – 1 0Given : A circle, 2x 2 2y 2 5 and a parabola, y 2 4 5 x.[JEE (Main)-2013]Statement–I : An equation of a common tangent to these curves is y x 5 .Statement–II : If the line, y mx 5(m 0) is their common tangent, then m satisfies m4 – 3m2 2 0.m(1) Statement-I is true, Statement-II is true; statement-II is a correct explanation for Statement-I.(2) Statement-I is true, Statement-II is true; statement-II is not a correct explanation for Statement-I.(3) Statement-I is true, Statement-II is false.(4) Statement-I is false, Statement-II is true.ANSWERP RE VIOU S Y EARS QU E STION SKEYE XE R CISE -5Q u e.123456789Ans342144232Page 11 of 14[A]

EXERCISE - 05 [B]1.( a ) If the line x – 1 0 is the directrix of the parabola y2 – kx 8 0, then one of the values of ‘k’ is:(A) 1/8(B) 8(C) 4(D) 1/42( b ) If x y k is normal to y 12x , then ‘k’ is (A) 32.(B) 9[JEE 2000 ( Screening) 1 1M](C) – 9(D) – 322( a ) The equation of the common tangent touching the circle (x – 3) y 9 and the parabola y2 4x abovethe x- axis is (A)3y 3x 1(B)(C)3y (x 3)(D)3y x 32( b ) The equation of the directrix of the parabola y 4y 4x 2 0 is (A) x – 13.(B) x 1[JEE 2001 ( Screening) 1 1M](C) x –3/2(D) x 3/2The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 4ax isanother parabola with directrix(A) x – a4.(B) x –a2(C) x 0(B) y 2x 1If a focal chord of the parabola yof the slope of this chord, are (A) {–1, 1}6.[JEE 2002 ( Screening), 3M](D) x The equation of the common tangent to the curves y2 8x and xy – 1 is (A) 3y 9x 25.228.a2[JEE 2002 (Scr), 3M](C) 2y x 8(D) y x 22 16x is a tangent to the circle (x – 6) y 2. then the set of possible values[JEE 2003 (Scr), 3M]1 (C) 2 , 2 (B) {–2, 2}1 (D) 2 , 2 Normals with slopes m 1, m 2, m 3 are drawn from the point P to the parabola y 2 4x. If locus of P withm1 m2 is a part of the parabola itself, find .7.3y (3 x 1)[JEE 2004 (Mains), 4M out of 60]2Two tangents are drawn from point (1, 4) to the parabola y 4x. Angles between tangents is (A) /6(B) /4(C) /3(D) /2[JEE 2004 (Screening), 3M]At any point P on the parabola y2 – 2y – 4x 5 0, a tangent is drawn which meets the directrix at Q. Find the1:1.[JEE 2004 (Mains), 4M out of 60]2Tangent to the curve y x2 6 at point P (1, 7) touches the circle x2 y2 16x 12y c 0 at a point Q.Then coordinate of Q is [JEE 2005 (Screening) 3M]locus of point R which divides QP externally in the ratio9.(A) (–6, 11)10.(B) (6, –11)(C) (–6, –7)(D) (–6, –11)The axis of a parabola is along the line y x and the distance of its vertex from origin is2 and that of originfrom its focus is 2 2 . If vertex and focus both lie in the first quadrant, then the equation of the parabola is [JEE 2006 (3M, –1M) out of 184]11.(A) (x y)2 (x – y – 2)(B) (x – y)2 (x y – 2)(C) (x – y)2 4(x y – 2)(D) (x – y)2 8(x y – 2)The equations of the common tangents to the parabola y x2 and y – x2 4x – 4 is/are(A) y 4(x – 1)12.(B) y 0(C) y –4(x – 1)(D) y –30x – 50[JEE 2006 , (5M, –1M) out of 184]Match the following[JEE 2006, (6M, 0M) out of 184]2Normals are drawn at points P, Q and R lying on the parabola y 4x which intersect at (3, 0). Then(i)(ii)(iii)(iv)Area of PQRRadius of circumcircle of PQRCentroid of PQRCircumcentre of PQR(A)(B)(C)(D)Page 12 of 1425/ 2(5/2, 0)(2/3, 0)

13 to 15 are based on this paragraph[JEE 2006 (5M, –2M) each, out of 184]Let ABCD be a square of side length 2 units. C2 is the circle through vertices A, B, C, D and C1 is the circletouching all the sides of the square ABCD. L is a line through A.13.14.PA 2 PB 2 PC 2 PD 2If P is a point on C1 and Q in another point on C2, thenis equal to QA 2 QB 2 QC 2 QD 2(A) 0.75(B) 1.25(C) 1(D) 0.5A circle touches the line L and circle C1 externally such that both the circles are on the same side of the line,then the locus of centre of the circle is (A) ellipse15.(B) hyperbola(C) parabola(D) pair of straight lineA line M through A is drawn parallel to BD. Point S moves such that its distances from the line BD and the vertexA are equal. If locus of S cuts M at T2 and T3 and AC at T1 then area of T1T2T3 is(A) 1/2 sq. units(B) 2/3 sq. units(C) 1 sq. units(D) 2 sq. units16 to 18 are based on this paragraphConsider the circle x 2 y 2 9 and the parabola y 2 8x. They intersect at P and Q in the first and thefourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents tothe parabola at P and Q intersect the x-axis at S.1 6 . The ratio of the areas of the triangle PQS and PQR is :

The equation of the parabola with point A as vertex, 2L as the length of the latus rectum and the axis at right angles to that of the given curve is - (A) x. 2 4x 8y – 4 0 (B) x 4x – 8y 12 0 (C) x. 2 4x 8y 12 0 (D) x 8x – 4y 8 0 3. The parametric coord