Title Conics- Daily-10 MCQ (Home Practice)-With Solve.pdf ✅

1 Varsity Daily-10 [Home Practice (Solve Sheet)] wm‡jevm: KwbK c~Y©gvb: 30 †b‡MwUf gvK©: 0.25 mgq: 20 wgwbU 1. 16x2 + 25y2 = 400 Dce„‡Ëi civwgwZK ̄’vbv1⁄4 †KvbwU? [What are the parametric coordinates of the ellipse 16x2 + 25y2 = 400?] (5cos,4sin) (5sin, 4cos) (4cos, 5sin) (4sin, 5cos) DËi: (5cos,4sin) e ̈vL ̈v: 16x2 + 25y2 = 400  x 2 25 + y 2 16 = 1 [Dfqc‡ÿ 400 Øviv fvM K‡i]  x 2 5 2 + y 2 4 2 = 1; †hLv‡b a = 5, b = 4 Avgiv Rvwb, x 2 a 2 + y 2 b 2 = 1 Dce„‡Ëi civwgwZK ̄’vbv1⁄4 x = acos, y = bsin  x 2 5 2 + y 2 4 2 = 1 Dce„‡Ëi civwgwZK ̄’vbv1⁄4 (5cos, 4sin) 2. y 2 = 8px cive„ËwU (4, – 8) we›`y w`‡q AwZμg K‡i| cive„ËwUi Dc‡K‡›`ai ̄’vbv1⁄4 n‡eÑ [If the parabola y2 = 8px passes through the point (4, - 8), what are the coordinates of its focus?] (0, 4) (4, 0) (16, 0) (0, 16) DËi: (4, 0) e ̈vL ̈v: y 2 = 8px cive„ËwU (4, – 8) we›`y w`‡q hvq|  (– 8)2 = 8p(4)  64 = 32p  p = 2 y 2 = 8  2x = 16x = 4  4  x myZivs, Dc‡K‡›`ai ̄’vbv1⁄4 n‡e (a, 0)  (4, 0) 3. y 2 = – 4ax cive„‡Ëi wbqvg‡Ki mgxKiY †KvbwU? [What is the equation of the directrix of the parabola y2 = – 4ax?] x + a = 0 y + a = 0 x – a = 0 y – a = 0 DËi: x – a = 0 e ̈vL ̈v: y 2 = – 4ax  y 2 = 4(– a)x y 2 = 4ax cive„‡Ëi wbqvg‡Ki mgxKiY, x = – a y 2 = – 4ax cive„‡Ëi wbqvg‡Ki mgxKiY x = – (– a)  x – a = 0 4. y 2 – 6x + 4y + 10 = 0 cive„‡Ëi A‡ÿi mgxKiY †KvbwU? [What is the equation of the axis of the parabola y2 – 6x + 4y + 10 = 0?] x + 1 = 0 x = 1 x – 2 = 0 y + 2 = 0 DËi: y + 2 = 0 e ̈vL ̈v: y 2 – 6x + 4y + 10 = 0  y 2 + 4y = 6x – 10  y 2 + 4y + 4 = 6x – 10 + 4  (y + 2)2 = 6x – 6  (y + 2)2 = 6(x – 1)  (y + 2)2 = 4  3 2  (x – 1)  kxl©we›`yi ̄’vbv1⁄4 (x – 1, y + 2) = (0, 0) x, y–  Dc‡K‡›`ai ̄’vbv1⁄4, (x – 1, y + 2) = (a, 0) x – 1 = 3 2  x = 3 2 + 1  x = 5 2 y + 2 = 0  y = – 2  Dc‡K‡›`ai ̄’vbv1⁄4     5 2  – 2 Aÿ‡iLv (1, – 2) Ges     5 2  – 2 we›`yMvgx|  Aÿ‡iLvi mgxKiY, x – 1 1 – 5 2 = y + 2 – 2 + 2  (y + 2)    1 – 5 2 = 0  y + 2 = 0 Shortcut: cive„‡Ëi Av`k© mgxKi‡Y †h As‡ki Dci eM© _vK‡e †mUv = 0 emv‡j Aÿ‡iLvi cvIqv hv‡e|  y + 2 = 0 Aÿ‡iLvi mgxKiY 5. y 2 9 – x 2 4 = 1 Awae„‡Ëi †dvKvm؇qi `~iZ¡ KZ GKK? [What is the distance between the foci of the hyperbola y 2 9 – x 2 4 = 1?] 13 2 2 13 4 13 3 13 DËi: 2 13 e ̈vL ̈v: y 2 b 2 – x 2 a 2 = 1 Awae„‡Ëi †dvKvm؇qi `~iZ¡ 2be x 2 a 2 – y 2 b 2 = 1 Awae„‡Ëi †dvKvm؇qi `~iZ¡ 2ae  y 2 3 2 – x 2 2 2 = 1 Awae„‡Ëi †dvKvm؇qi `~iZ¡ = 2be = 2  a 2 + b2 = 2  9 + 4
2 = 2 13 GKK [Awae„‡Ëi ae= be = a 2 + b2 ] 6. †Kv‡bv Dce„‡Ëi Dc‡Kw›`aK j¤^ Dce„ËwUi e„nr A‡ÿi A‡a©K n‡j Dr‡Kw›`aKZv KZ? [If the latus rectum of an ellipse is half of its major axis, what is its eccentricity?] 1 2 2 1 2 2 DËi: 1 2 e ̈vL ̈v: g‡b Kwi, Dce„‡Ëi Dc‡Kw›`aK j‡¤^i •`N© ̈ = 2b2 a | †hLv‡b, a > b e„nr A‡ÿi •`N© ̈ = 2a cÖkœg‡Z, 2b2 a = 1 2  2a  4b2 = 2a2  b 2 a 2 = 2 4 = 1 2  Dr‡Kw›`aKZv, e = 1 – b 2 a 2 = 1 – 1 2 = 1 2 = 1 2 7. y = mx + c mij‡iLvwU y 2 = 8x cive„ˇK ̄úk© Ki‡j, ̄úk© we›`yi ̄’vbv1⁄4 KZ? [If the line y = mx + c is tangent to the parabola y2 = 8x, what are the coordinates of the point of tangency?] (4m, 2m2 )     2 m  4 m 2     4 m  2 m 2     2 m 2  4 m DËi:     2 m 2  4 m e ̈vL ̈v: y = mx + c †iLv y 2 = 4ax cive„‡Ëi ̄úk©K n‡j, ̄úk©we›`y     a m 2  2a m cive„ËwU, y 2 = 8x = 4  2  x [GLv‡b, a = 2]  ̄úk©we›`y     2 m 2  4 m 8. x 2 + 4xy + 4y2 + 2x + 4y + 1 = 0 Kx‡mi mgxKiY wb‡`©k K‡i? [What does the equation x2 + 4xy + 4y2 + 2x + 4y + 1 = 0 represent?] e„Ë (circle) cive„Ë (parabola) Dce„Ë (ellipse) hyMj mij‡iLv (pair of straight lines) DËi: hyMj mij‡iLv (pair of straight lines) e ̈vL ̈v: x 2 + 4xy + 4y2 + 2x + 4y + 1 = 0 mgxKi‡Y, a = 1, b = 4, h = 2, g = 1, f = 2, c = 1  =       a h g h b f g f c =       1 2 1 2 4 2 1 2 1 1g mvwi I 3q mvwi GKB|   = 0 ZvB mgxKiYwU †Rvov mij‡iLv n‡e| 9. ÿz`a A‡ÿi •`‡N© ̈i wZb ̧Y •`‡N© ̈i e„nr Aÿ wewkó Dce„‡Ëi Dr‡Kw›`aKZv Gi Rb ̈ †KvbwU mZ ̈? [What are the vertices of the hyperbola x2 - 3y2 - 2x = 8?] 2 3 2 3 2 2 3 4 3 DËi: 2 2 3 e ̈vL ̈v: hw` a > b e„nr A‡ÿi •`N© ̈ = 2a ÿz`a A‡ÿi •`N© ̈ = 2b cÖkœg‡Z, 2a = 3  2b  a = 3b  b a = 1 3  b 2 a 2 = 1 9  Dr‡Kw›`aKZv e = 1 – b 2 a 2 = 1 – 1 9 = 8 3 = 2 2 3 10. x 2 – 3y2 – 2x = 8 Awae„‡Ëi kxl©we›`yØqÑ [If the length of the major axis of an ellipse is three times the length of its minor axis, what is the eccentricity of the ellipse?] (0, – 2), (0, 4) (– 4, 0), (2, 0) (– 2, 0), (4, 0) (0, – 4), (0, – 1) DËi: (– 2, 0), (4, 0) e ̈vL ̈v: x 2 – 3y2 – 2x = 8  x 2 – 2x + 1 – 3y2 = 8 + 1  (x – 1)2 – 3y2 = 9  (x – 1) 2 9 – y 2 3 = 1  kxl©we›`yØq  (x – 1, y) = ( a, 0)  x – 1 =  3  x = – 2, 4  kxl©we›`yØq (– 2, 0), (4, 0) 11. y = 2x + c †iLvwU x 2 4 + y 2 3 = 1 Dce„‡Ëi ̄úk©K n‡j, c Gi gvb KZ? [If the line y = 2x + c is tangent to the ellipse x 2 4 + y 2 3 = 1, what is the value of c?] 19 – 19  19  7 DËi:  19 e ̈vL ̈v: y = 2x + c †iLvq m = 2
3 y = mx + c †iLvwU x 2 a 2 + y 2 b 2 = 1 Dce„‡Ëi ̄úk©K n‡j, c 2 = a2m 2 + b2 = 4  4 + 3 [a2 = 4 Ges b 2 = 3] = 19  c =  19 12. 4x2 + y2 = 2 Dce„ËwUi ÿz`a I e„nr A‡ÿi •`N© ̈ h_vμ‡gÑ [What are the lengths of the minor and major axes of the ellipse 4x2 + y2 = 2?] 2 2 Ges 2 2 Ges 2 2 4 I 2 2 I 4 DËi: 2 Ges 2 2 e ̈vL ̈v: 4x2 + y2 = 2  2x2 + y 2 2 = 1  x 2 1 2 + y 2 2 = 1  x 2     1 2 2 + y 2 ( 2) 2 = 1 [b > a]  ÿy`a A‡ÿi •`N© ̈ = 2a = 2  1 2 = 2 GKK e„nr A‡ÿi •`N© ̈ = 2b = 2  2 GKK 13. (x – 3)2 = – 4(y – 4) cive„ËwUi Dc‡Kw›`aK j‡¤^i mgxKiYÑ [What is the equation of the latus rectum of the parabola (x - 3)2 = -4(y - 4)?] x – 3 = 0 y + 3 = 0 x + 3 = 0 y – 3 = 0 DËi: y – 3 = 0 e ̈vL ̈v: (x – 3)2 = 4  (– 1)  (y – 4) Dc‡Kw›`aK j‡¤^i mgxKiY, y – 4 = – 1  y – 3 = 0 14. y = – (x – 1)2 cive„‡Ëi †jLwPÎ wb‡Pi †KvbwU? [Which graph represents the equation y = -(x - 1)2?] 1 –1 –1 1 DËi: 1 e ̈vL ̈v: cive„Ë, y = – (x – 1)2  (x – 1)2 = – y cive„‡Ëi kxl© (1, 0) Ges x 2 = – 4ay AvKv‡ii| 15. 9x2 + 5y2 = 45 Dce„‡Ëi Dc‡K›`a؇qi ga ̈eZx© `~iZ¡ KZ? [What is the distance between the foci of the ellipse 9x2 + 5y2 = 45?] 8 GKK (8 unit) 6 GKK (6 unit) 4 GKK (4 unit) 3 GKK (3 unit) DËi: 4 GKK (4 unit) e ̈vL ̈v: 9x2 + 5y2 = 45  x 2 5 + y 2 9 = 1  x 2 ( 5) 2 + y 2 3 2 = 1 [b > a]  Dc‡K›`a؇qi ga ̈eZx© `~iZ¡ = 2be = 2  b 2 – a 2 = 2  9 – 5 = 2  2 = 4 GKK 16. x 2 5 2 + y 2 9 = 1 Dce„‡Ëi (1, 2) we›`y‡Z ̄úk©‡Ki mgxKiY †KvbwU? [What is the equation of the tangent to the ellipse x 2 5 2 + y 2 9 = 1 at the point (1, 2)?] x 25 + y 9 = 1 2x 25 + y 9 = 1 x 25 + 2y 9 = 1 x 9 + 2y 3 = 1 DËi: x 25 + 2y 9 = 1 e ̈vL ̈v: x 2 a 2 + y 2 b 2 = 1 Dce„‡Ëi (x1, y1) we›`y‡Z ̄úk©‡Ki mgxKiY, xx1 a 2 + yy1 b 2 = 1  x 2 5 2 + y 2 3 2 = 1 Dce„‡Ëi (1, 2) we›`y‡Z ̄úk©‡Ki mgxKiY, x  1 25 + y  2 9 = 1  x 25 + 2y 9 = 1 17. 4x2 – 9y2 – 8x + 18y – 41 = 0 KwY‡Ki AmxgZU؇qi Xv‡ji ̧Ydj †KvbwU? [What is the product of the slopes of the asymptotes of the conic 4x2 - 9y2 - 8x + 18y - 41 = 0?] 4 9 2 3 – 4 9 – 9 4 DËi: – 4 9 e ̈vL ̈v: 4x2 – 9y2 – 8x + 18y – 41 = 0  4(x2 – 2x + 1) – 9(y 2 – 2y + 1) = 41 + 4 – 9  4(x – 1)2 – 9(y – 1)2 = 36  (x – 1) 2 9 – (y – 1) 2 4 = 1 x 2 a 2 – y 2 b 2 Awae„‡Ëi AmxgZ‡Ui mgxKiY, y –  =  b a (x – ) (y – 1) =  2 3 (x – 1)  Xv‡ji ̧Ydj = 2 3  – 2 3 = – 4 9 18. 4x2 + 9y2 = 36 Dce„ËwUi †ÿÎdj KZ? [What is the area of the ellipse 4x2 + 9y2 = 36?] 36 eM©GKK (36π square units) 12 eM©GKK (12π square units)
4 6 eM©GKK (6π square units) 8 eM©GKK (8π square units) DËi: 6 eM©GKK (6π square units) e ̈vL ̈v: Dce„Ë, 4x2 + 9y2 = 36  x 2 9 + y 2 4 = 1  x 2 3 2 + y 2 2 2 = 1  Dce„‡Ëi †ÿÎdj = ab =   3  2 = 6 eM©GKK 19. 3(x – 1)2 + (2y – 1)2 = 12 – 4y + 1 mgxKiY Kx eY©bv K‡i? [What does the equation 3(x - 1)2 + (2y - 1)2 = 12 - 4y + 1 represent?] e„Ë (circle) cive„Ë (parabola) Dce„Ë (ellipse) Awae„Ë (hyperbola) DËi: Dce„Ë (ellipse) e ̈vL ̈v: 3(x – 1)2 + (2y – 1)2 = 12 – 4y + 1  3(x – 1)2 + 4y2 = 12  (x – 1) 2 2 2 + y 2 ( 3) 2 = 1; hv Dce„Ë wb‡`©k K‡i| 20. (y + 2)2 – 4y – 4 = 16x cive„‡Ë Dcwiw ̄’Z †Kvb we›`yi Dc‡Kw›`aK `~iZ¡ 6 ; H we›`yi ̄’vbv1⁄4 KZ? [If the latus rectum of the parabola (y + 2)2 - 4y - 4 = 16x is 6, what are the coordinates of the point on the parabola?] (2  4 2) (2   4 2) (2  – 4 2)  DËi: (2   4 2) e ̈vL ̈v: y 2 + 4y + 4 – 4y – 4 = 16x y 2 = 16x = 4.4x a = 4 SP = PM  6 = AZ + AN  6 = AS + AN P(x, y) X z M M A N S Y  6 = a + x  x = 2 y 2 = 4.4.2  y =  4 2 (x, y) = (2   4 2) 21. †Rbv‡ij AvwRR evsjv‡`k †mbv cÖavb ev›`iev‡b Rw1⁄2i †Mvwô‡K ai‡Z Awfhvb Pvjvb| D3 Awfh‡b Zv‡K GKwU U ̈v‡b‡j cÖ‡ek Ki‡Z nq| †mB U ̈v‡b‡j cÖ‡ek Ki‡Z †mbv cÖavb U ̈vsK wb‡q hvIqvi Rb ̈ Av‡`k †`b| KZ cÖk ̄Í Ges D”PZvi U ̈vsK Zv‡K wb‡Z n‡e? [General Aziz Bangladesh Army led an operation to catch the militant group in Bandarban. In the mission he has to enter a tunnel. The army chief ordered tanks to enter the tunnel. How wide and tall will the tank take him?] 3a, 4a a, 4a a, a 6a, a DËi: a, 4a e ̈vL ̈v: U ̈v‡bj cive„ËvKvi n‡q _v‡K| D”PZv = g~j we›`y A_©vr, U ̈v‡b‡ji D”PZv n‡Z f‚wgi `~iZ¡| = a cÖk ̄Í = Dc‡Kw›`aK j‡¤^i •`N© ̈ = 4a     3 5  0 22. GKwU Dce„‡Ëi Dc‡Kw›`aK j¤^ ÿz`a A‡ÿi A‡a©K n‡j Dr‡Kw›`aKZv KZ? [If the length of the latus rectum of an ellipse is half the length of its minor axis, what is its eccentricity?] cos  3 cos  6 cos  4 sin  6 DËi: cos  6 e ̈vL ̈v: 2b2 a = b 2b a = 1 b a = 1 2 e = 1 –     1 2 2 = 3 2 Avgiv Rvwb, cos  6 = 3 2 23. x + 2y – 1 = 0 mij‡iLvi Dci j¤^ Ges y 2 = 12x cive„‡Ëi ̄úk©‡Ki mgxKiY †KvbwU? [Which of the following is the equation of the normal to the line x + 2y – 1 = 0 at the point of intersection with the parabola y2 = 12x?] 2y – 4x – 3 = 0 2x – y + 1 = 0 x + 2y – 1 = 0 4x – 2y + 3 = 0 DËi: 2y – 4x – 3 = 0 e ̈vL ̈v: y 2 = 12x y 2 = 4  3  x a = 3 x + 2y – 1 = 0 Gi j¤^ †iLvq mgxKiY, 2x – y + c = 0 m = 2 c = a m = 3 2 y = 2x + 3 2 2y – 4x – 3 = 0 24. 5x2 + 30x + 2y + 59 = 0 cive„‡Ëi Aÿ‡iLvi mgxKiY †KvbwU? [What is the equation of the axis of the parabola 5x2 + 30x + 2y + 59 = 0?] x – 2 = 0 x + 3 = 0 x = 0 y + 5 = 0 DËi: x + 3 = 0 e ̈vL ̈v: 5(x2 + 6x) = – 2y – 59