1. If y 2 2x 2y + 5 = 0 is (A) a circle with centre (1, 1) (B) a parabola with vertex (1, 2) 9 (A) 0, (B) 4, (C) (4, 4) (D) a (C) c = am m.

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Beverley Richards
5 years ago
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1 SET I. If y x y + 5 = 0 is (A) circle with centre (, ) (B) prbol with vertex (, ) (C) prbol with directrix x = 3. The focus of the prbol x 8x + y + 7 = 0 is (D) prbol with directrix x = 9 9 (A) 0, (B) 4, (C) (4, 4) (D) 4, 3. Eqution of common tngent to the circle, x + y = 50 nd the prbol, y = 40x cn be (A) x + y - 0 = 0 (B) x - y + 0 = 0 (C) x y 0 = 0 4. The line y = mx + c touches the prbol x = 4y if (A) c = m (B) c = (C) c = m m (D) m 5. The length of the ltus rectum of the prbol 4y + x 0y + 7 = 0 is (A) 3 (B) 6 (C) (D) 9 6. In prbol semi-ltus rectum is the hrmonic men of the (A) segments of focl chord (B) segments of the directrix (C) segments of chord 7. If (t, t) is one end of focl chord of the prbol, y = 4x then the length of the focl chord will be : (A) t (B) t t t t t (C) t t t t 8. The eqution of the prbol with its vertex t (, ) nd focus t (3, ) is (A) (x 3) = 8(y ) (B) (y ) = 8(x ) (C) (y ) = 8(x 3) (D) (x ) = 8(y ) 9. Eqution of the tngent t ( 4, 4) on x = 4y is (A) x y + 4 = 0 (B) x + y 4 = 0 (C) x y = 0 (D) x + y + 4 = 0 0. The locus of the midpoint of the line segment joining the focus to moving point on the prbol y = 4x is nother prbol with directrix

2 (A) x = (B) x = (C) x = 0 (D). The point of intersection of the tngents to the prbol t the points t nd t is (A) ( t t, (t + t ) ) (B) ( t t, (t + t ) ) (C) ( t t, (t + t ) ) x. The tngent drwn t ny point P to the prbol y = 4x meets the directrix t the point K. Then the ngle which KP subtends t the focus is (A) 30 0 (B) 45 0 (C) 60 0 (D) The eqution of the prbol whose focus is the point (0, 0) nd the tngent t the vertex is x y + = 0 is (A) x + y xy 4x + 4y 4 = 0 (B) x + y xy + 4x 4y 4 = 0 (C) x + y + xy 4x + 4y 4 = 0 (D) x + y + xy 4x 4y + 4 = 0 4. If (x, y ), (x, y ), (x 3, y 3 ) be three points on the prbol y = 4x, the normls t which meet in point, then (A) y + y = 0 (B) x + x + x 3 = 0 (C) y + y = (D) x + x + x 3 = 4 5. The norml t point on y = 4x psses through (5, 0). There re three such normls one of which is the xis. The feet of the three normls from tringle whose centroid is (A) (, 0) (B) (0, ) (C), (D) (5, 0) 6. The locus of the point of intersection of the perpendiculr tngents to the prbol x = 4y is (A) y = (B) y = (C) x = (D) x = 7. The norml t the point P(p, p) meets the prbol y = 4x gin t Q(q, q) such tht the lines joining the origin to P nd Q re t right ngle: Then (A) p = (B) q = (C) p = q (D) q = p 8. If x + y + k = 0 is norml to the prbol y = 8x, then the vlue of k is (A) 6 (B) 8 (C) 4 (D) 4 9. The ngle between the tngents drwn from the origin to the prbol y = 4(x ) is (A) 90 (B) 45 (C) 60 (D) tn 0. The prmetric coordintes of ny point on the prbol y = x cn be (A) (sec, sec ) (B) (sin, sin ) (C) (cos, cos )

3 SET II. If the normls t points t nd t meet on the prbol, then (A) t t = (B) t = t (C) t t = t. Which one of the following equtions represented prmetriclly, represents eqution to prbolic profile? (A) x = 3 cos t ; y = 4 sin t (B) x = cos t ; y = 4 cos t (C) x = tn t ; y = sec t (D) x = sin t ; y = sin t + cos t 3. The condition tht the two tngents to the prbol y = 4x become norml to the circle x + y x by + c = 0 is given by (A) > 4b (B) b > (C) > b (D) b > 4 4. The locus of the middle points of focl chords of prbol is (A) y = (x ) (B) y = (x + ) (C) x = (y ) (D) x > (y + ) 5. The norml chord of prbol y = 4x t (x, x ) subtends right ngle t the (A) focus (B) vertex (C) end of the ltusrectum 6. If the chord y = mx + c subtends right ngle t the vertex of the prbol y = 4x, then the vlue of c is (A) 4m (B) 4 m (C) m (D) m 7. If P( 3, ) is one end of the focl chord PQ of the prbol y + 4x + 4y = 0, then the slope of the norml t Q is (A) (B) (C) (D) 8. The ngle between tngents to the prbol y = 4x t the points where it intersects with the line x y = 0 is (A) 3 (B) 4 (C) 6 (D) 9. If the lines (y b) = m (x + ) nd (y b) = m (x + ) re the tngents of y = 4x then (A) m + m = 0 (B) m m = (C) m m = (D) m + m = 0. The prbol y = 4x nd the circle (x 6) + y = r will hve no common tngent if r is equl to (A) r > 0 (B) r < 0 (C) r > 8 (D) r ( 0, 8. Prbols y = 4(x c ) nd x = 4(y c ), where c nd c re vrible, re such tht they touch ech other. Locus of their point of contct is (A) xy = (B) xy = 4 (C) xy =. Minimum distnce between the curve y = 4x nd x + y x + 3 = 0 is equl to

4 (A) (B) 6 5 (C) 5 (D) The locus of the foot of the perpendiculr from the focus upon tngent to the prbol y = 4x is (A) the directrix (B) tngent t the vertex (C) x = 4. If (x r, y r ); r =,, 3, 4 be the points of intersection of the prbol y = 4x nd the circle x + y + gx + fy + c = 0, then (A) y + y + y 4 = 0 (B) y + y y 3 y 4 = 0 (C) y y y 4 = 0 (D) y y y 3 + y 4 = 0 5. The length of focl chord of the prbol y = 4x mking n ngle with the xis of the prbol is (A) 4 cosec (B) 4 sec (C) cosec 6. Three normls to the prbol y = x re drwn through point (C, 0) then (A) C = 4 (B) C = (C) C > 7. The tringle PQR of re 'A' is inscribed in the prbol y = 4x such tht the vertex P lies t the vertex of the prbol nd the bse QR is focl chord. The modulus of the difference of the ordintes of the points Q nd R is : (A) A (B) A (C) A (D) 4A 8. The tngent nd norml t the point P(t, t) to the prbol y = 4x meet the x-xis in T nd G respectively, then the ngle t which the tngent t P to the prbol is inclined to the tngent t P to the prbol is inclined to the tngent t P to the circle through P, T, G is (A) tn - (t ) (B) cot - (t ) (C) tn - (t) (D) cot - (t) 9. From n externl point P, pir of tngent lines re drwn to the prbol, y = 4x. If & re the inclintions of these tngents with the xis of x such tht, + =, then the locus of P is : 4 (A) x - y + = 0 (B) x + y - = 0 (C) x - y - = 0 (D) x + y + = 0 0. Length of the chord of contct of the pir of tngents drwn from on the point (x, y ) the prbol, y = 4x is : (A) y 4x y 4 (B) y x y 4 4 (C) y 4x y 4 (D) y 4x y 4

5 SET III. Eqution x x y + 5 = 0 represents (A) circle with centre (, ) (B) prbol with vertex (, ) (C) prbol with directrix y = 5/ (D) prbol with directrix y = /3. The normls to the prbol y = 4x from the point (5, ) re (A) y = x 3 (B) y = x + (C) y = 3x + 33 (D) y = x The eqution of the lines joining the vertex of the prbol y = 6x to the points on it whose bsciss is 4, is (A) y ± x = 0 (B) y ± x = 0 (C) x ± y = 0 (D) x ± y = 0 4. The eqution of the tngent to the prbol y = 9x which goes through the point (4, 0) is (A) x + 4y + = 0 (B) 9x + 4y + 4 = 0 (C) x 4y + 36 = 0 (D) 9x 4y + 4 = 0 5. Consider the eqution of prbol y + 4x = 0, where > 0. Which of the following is flse? (A) tngent t the vertex is x = 0 (B) directrix of the prbol is x = 0 (C) vertex of the prbol is t the origin (D) focus of the prbol is t (, 0) Question bsed on write-up Normlly, the vrious propositions you study, e.g. eqution of tngent, norml, chord, focl chord, formul for focl distnce etc, re derived for the prbol y = 4x. However, ll the results with slight trnsformtion re vlid for ny prbol. Suppose we represent the eqution of prbol y 4x = 0 by S (x, y, ) = 0 nd ny eqution derived for this prbol by P(x, y, ) = 0. Now, if the given prbol is y = 4x, y + 4x = 0 we cn write, if S(x, y, ) = 0, so the corresponding eqution of P will be P(x, y, ) = 0. Similrly for x = 4y cn be written s S(x, y, ) nd corresponding trnsformtion is P(x, y, ) (i.e. interchnge x nd y). 6. The focl distnce of the point (x, y) on the prbol x 8x + 6y = 0 is (A) y 4 (B) y 5 (C) y (D) x 4 7. Normls re drwn from the point (7, 4) to the prbol x 8x 6y = 0. The slopes of these normls re 3 3 (A) (B) (C) (D) 8. The coordintes of the feet of the normls obtined in previous problem re (A) (0, 8) (B) (4, 3) (C) (6, 8) (D) ( 8, 4) 9. The points on the xis of the prbol x + x + 4y + 3 = 0 from the where three distinct normls cn be drwn re given by (A) (, k), k (, ) (B) (, 6) (C) (, k), k (, 5) (D) (, ) 0. The line x cos ysin p touches the prbol x + 4(y + ) = 0, if

6 (A) = p sec (B) cos = p sin (C) cos + p sin = 0 (D) tn = p sec Three normls cn be drwn from point (x,y ) to the prbol y = 4x. The points where these normls meet the prbol re clled the feet of the normls or conorml points. The sum of the slopes of the normls is zero nd the sum of the ordintes of the feet of the normls is lso zero. m + m + m 3 = 0 y + y = - (m + m + m 3 ) = 0 m m + m m 3 + m 3 m = m m m 3 = x y, where m, m, m 3 re slopes nd y, y, y 3 re ordintes.. The locus of the point of intersection of the three normls to the prbol y = 4x, two of which re inclined t right ngles to ech other is (A) y{y + (3 + x)}= 0 (B) y{y + (3 x)}= 0 (C) y{y (3 x)}= 0. The normls t three points P, Q, R of the prbol y = 4x meet in (h, k). The centroid of tringle PQR lies on (A) x = 0 (B) y = 0 (C) x = 3. The number of distinct normls tht cn be drwn from, to the prbol y = 4x is 4 4 (A) (B) (C) 3 4. Three normls to the prbol y = x re drwn through point (C, 0), then (A) C = 4 (B) C = (C) C > 5. If the normls from ny point to the prbol x = 4y cuts the line y = in points whose bsciss re in AP, then the slopes of the tngents t the three conorml points re in (A) G.P (B) A.P. (C) H.P. 6. The locus of the points such tht two of the three normls from them to the prbol y = 4x coincide is : (A) 7y + 4 (x + ) 3 = 0 (B) 7y + 4 (x ) 3 = 0 (C) 7y = 4 (x ) 3 7. Assertion : Solpe of tngents drwn from (4, 0) to prbol y = 9x re Reson : Every prbol is symmetric bout its directrix. (A) both Assertion nd Reson re true nd Assertion is the correct explntion of Reson,

7 8. True/Flse : (i) (B) (C) (D) both Assertion nd Reson re true nd Assertion is not the correct explntion of Reson Assertion is true but Reson is flse Assertion is flse but Reson is true The eqution of the prbol whose focus is t the origin is y = 4(x + ). (ii) The locus of the mid points of the chords of the prbol y = 4x which pss through the vertex, is the prbol y = x (iii) The focus of the prbol x + 8y = 0 is t (0, ) (iv) (v) The line y = mx + c is tngent to the prbol y = 4(x + ), then c m The points x = t, y = t re lies on the prbol x = 4y. m 9. Fill in the blnks : (i) The eqution of the prbol whose focus is the point (, 3) nd directrix is the line x 4y + 3 = 0 is...nd the length of its ltus rectum is... (ii) For the prbol y + 4x 6y + 3 = 0, the vertex is..., focus is... directrix is...l.r. is... (iii) The length of the ltus rectum of the prbol x 4x 8y + = 0 is... (iv) The focus of the prbol y = x + x is... (v) The vertex of the prbol (y ) = 6(x ) is Mtch the column Column I Column II () The ltus rectum of the prbol y = 5x + 4y + is (P) x = 3 (b) Two perpendiculr tngents to y = 4x lwys intersect on the line (Q) 0 (c) The eqution of the directrix of the prbol y + 4y + 4x + = 0 is (R) 3 4 (d) The number of tngent(s) to the prbol y = 8x through (, ) is (S) x = (e) If two different tngents of y = 4x re the normls to x = 4by then b is less thn (T) 5 (f) Minimum distnce between the curves y = x nd x = y is (U)

8 LEVEL I ANSWER 4 4. y = x + 3x + 4, LR = unit 4.,,, x + y = unit 9. y = -4x + 7, y = 3x y = 4(x - 8) LEVEL II 4 4.,, (4m, 4m) m m SET I. C. C 3. B 4. C 5. C 6. A 7. A 8. C 9. A 0. C. A. D 3. C 4. A 5. A 6. B 7. A 8. D 9. A 0. D SET II. C. B 3. D 4. A 5. A 6. A 7. A 8. D 9. C 0. B. B. C 3. B 4. A 5. A 6. C 7. C 8. C 9. C 0. C SET III. BC. AB 3. BC 4. CD 5. BD 6. B 7. C 8. C 9. C 0. B. B. B 3. C 4. C 5. C 6. C 7. C 8. (i) T (ii) T (iii) F (iv) T (v) T 9. (i) 6x + y + 8xy 74x 78y +, 4 7 (ii) (, 3), (, 3), x = 0, 4 (iii) 8 (iv), 0 4 (v) (, ) 0. -T, b-s, c-p, d-q, e-u, f-r

PARABOLA EXERCISE 3(B)

PARABOLA EXERCISE 3(B) PARABOLA EXERCISE (B). Find eqution of the tngent nd norml to the prbol y = 6x t the positive end of the ltus rectum. Eqution of prbol y = 6x 4 = 6 = / Positive end of the Ltus rectum is(, ) =, Eqution