d) Find the equation of the circle whose extremities of a diameter are (1,2) and (4,5).

Transcription

1 ` KUKATPALLY CENTRE IPE MAT IIB Imortant Questions

2 a) Find the equation of the circle whose centre is (-, ) and which asses through (,6) b) Find the equation of the circle assing through (,) and concentric with the circle = c) If the circle + + a+ b- = has the centre at (,) then find a, b and its radius d) Find the equation of the circle whose etremities of a diameter are (,) and (,) e) Find the other end of the diameter of the circle = if one end of it is (, ) f) Obtain the arametric equation of the circle reresented b = g) Locate the osition of the oint (, ) with resect to the circle = h) Find the length of the tangent from (, ) to the circle = i) Find the ower of the oint P with resect to the circle S = when P= (, - 6) and S= j) Find the equation of the olar of (, ) with resect to the circle = k) Show that (, -) and (, -6) are conjugate with resect to the circle + - = l) If (, k) and (, ) are conjugate oints with resect to the circle + = 7 then find k a) Find the equation of the shere whose centre is (, -, ) and radius is b) Find the centre and radius of the shere + + z z= c) Find the equation of the shere that asses through the oint (,, -) and having its centre at (, 8, ) d) Find the equation of the shere having (,, ) and (-, 6, -7) as the etremities of one of its diameters e) If (,, ) is one end of a diameter of the shere z 6 z =, then find the coordinates of the other end of the diameter a) Find the equation of the arabola whose verte is (, -) and focus is (, ) b) Find the coordinates of the oints on the arabola = whose focal distance is c) If (/, ) is one etremit of a focal chord of the arabola = 8 Find the coordinates of the other etremit d) Show that the straight line 7+ 6= is a tangent to the arabola = and find the oint of contact e) Find the value of k if the line = + k is a tangent to the arabola = 6 f) Find the value of k if oints (,), (k,-) are conjugate with resect to the arabola = 8

3 a) Find the equation of the ellise referred to its major and minor aes as the coordinate aes, -resectivel with latus rectum of length and distance between foci b) If the length of the latus rectum is equal to half of its minor ais of an ellise in the standard form, then find the eccentricit of the ellise c) S and T are the foci of an ellise and B is one end of the minor ais If STB is an equilateral triangle, then find the eccentricit of the ellise d) Find the equation of the ellise in the standard form such that distance between foci is 8 and distance between directrices is e) Find the eccentricit of the ellise (in standard form), if its length of the latus rectum is equal to half of its major ais f) Find the value of k if + + k= is a tangent to the ellise + = a) If e, e are the eccentricities of a herbola and its conjugate herbola rove that + = e e b) Find the equations of the herbola whose foci are ( ±,), the transverse ais is of length 8 c) If - + k= is a tangent to - = find the value of k d) Find the roduct of lengths of the erendiculars from an oint on the herbola - = to its asmtotes 6 9 e) If the eccentricit of a herbola is, then find the eccentricit of its conjugate herbola 6 a) Find the th derivative of + c) Find the nth derivative of f( ) log( 8 6 7) b b) Find the rd derivative of e cos = for all d) Suose f( ) = a+ " ¹ Then show that f ( ) f ( ) f ( ) e) Find the nth derivative of g) If = ( - )( - ), find n 7 Evaluate the following: >- " + ' - = sin f) Find the nth derivative of sin sin h) Find the nth derivative of log( 9) - a) cot d b) æ 6 ö - d ç è+ ø c) ( a -b ) d d) sec cosec d e) + cos d -cos f) - cos d g) cosh+ sinh d h) d + cos i) cos sind j) sin d 6 cos k) sin d l) d ( + ) +

4 m) d n) -9 d + o) 9 - d ) 6- d q) d 6 - r) sin d s) d + e t) e ( sin+ cos) d u) ( tan + sec ) e d v) 8 Evaluate the following definite integrals æ logö e + d ç è ø on ( ), w) d + a) d + b) - d c) d + d) d + e) sin d sin + cos f) sin -cos d sin + cos g) sin d h) cos d i) 9 a) Find the area under the curve f( ) = sin in [ ], æ - ö log ç d çè + ø - b) Find the areounded b the arabola =, the X -ais and the lines =--, = c) Find the area cut off between the line = and the arabola = - + d) Find the area of one of the curvilinear triangles bounded b = sin, = cos and X-ais e) Find the areounded between the curves =, = f) Find the area of the region enclosed b the curves = sin, = cos, = = g) Find the area of the region enclosed b the curves = -, = æ d ö d a) Find the order and degree of + + = ç çè d ø d and 6 é ù d ædö + = 6 êd ç èdø ú ë û b) Find the differential equation corresonding to = Acos + Bsin (A, B are arameters) c) Solve the following differential equations i) d + = d + e + d+ + d= ii) ( ) ( ) iii) d - - = e + e iv) d + d+ + d= v) d d + e + d+ + d= = vi) ( ) ( )

5 MARKS QUESTIONS a) If a oint P is moving such that the lengths of tangents drawn from P to the circles = and equation of the locus of P = are in the ratio : then find the b) Find the length of the chord interceted b the circle - + = = on the line c) Find the equation of the tangent to tangent arallel to it = at (, -) Also find the equation of d) Find the area of the triangle formed b the tangent at P(, ) to the circle coordinate aes where ¹ + = a with the e) Find the equation of the circle with centre (,) and touching the line - + = f) Find the equation of the circle with centre (-, ) and touching -ais g) Find the equation of the circle with centre (-, ) cutting a chord length units on + + = h) Find the condition that the tangents drawn from the eterior oint (g, f) to S= + + g+ f+ c= are erendicular to each other i) Find the ole of + + = with resect to the circle = a) Find the equation of the arabola assing through the oints (-, ), (, -) and (, ) and having its ais arallel to the -ais b) A double ordinate of the curve ends are at right angles = a is of length 8a Prove that the line from the verte to its c) Find the equation of the arabola whose ais is arallel to -ais and which asses through the oints (, ), (-, ) and (-, ) d) Find the equation of the arabola whose focus is (-, ) and directri is the line + - = Also find the length of the latus rectum and the equation of the ais of the arabola e) If l+ m+ n= is a normal to the arabola = a, then show that al + alm + nm =

6 f) Show that the equation of common tangents to the circle are =± ( + a) + = a and the arabola = 8a g) Prove that the oles of normal chords of the arabola ( + a) + a = = a lie on the curve h) Prove that the oles of tangents to the arabola lie on a arabola = a with resect to the arabola = b a) Find the length of major ais, minor ais, latus rectum, eccentricit, coordinates of centre, foci and the equations of directrices of the following ellise = b) Find the equation of tangent and normal to the ellise rectum in the first quadrant = at the end of the latus c) If the normal at one end of a latus rectum of the ellise + = asses through one end of the minor ais, then show that e + e = [e is the eccentricit of the ellise] d) Find the equations to the tangents to the ellise find the angle between these tangents + = drawn from the oint (, ) and also e) Find the equation of the tangents to the ellise + = 8 which are (i) arallel to - - = (ii) erendicular to + + = (iii) which makes an angle with -ais f) Show that the oles of the tangents to the circle + = wrt the circle + = a lies on the curves a + b = a g) Show that the oles of the tangents to the circle + = a + b wrt the ellise + = lies on + = a + b h) Find the centre, foci, eccentricit, equation of the directrices, length of the latus rectum of the following herbola = i) Find the equation to the herbola whose foci are (, ) and (8, ) and eccentricit is

7 j) Find the equations of the tangents to the herbola erendicular to the line + = - = which are (i) arallel (ii) k) Find the equations of tangents drawn to the herbola 6 - = through (,) - l) Show that the feet of the erendiculars drawn from foci to an tangent of the herbola - = lie on the auiliar circle of the herbola æ ö a) Find the area of the triangle formed b the oints with olar coordinates ç, çè 6 ø, æ, ö ç çè ø, æ, ö ç çè ø b) Show that the oints with olar coordinates (, ), (, ) and (, 6) form an equilaterial triangle æ ö c) Find the olar equation of the straight line joining the oints ç, çè ø and æ, ö ç çè ø d) Find the olar equation of the line assing through (, ) and (i) arallel (ii) erendicular to the line cosq+ sinq= r e) Find the olar coordinates of the oint of intersection of the lines cosq+ cos( q- ) = r and cosq+ cos( q+ ) = r f) Polar equation of a conic in the standard form is l e cos r = + q ( l is semilatus rectum, e is eccentricit) g) If PSQ is a chord assing through the focus S of a conic and l is semi latus rectum, show that + = SP SQ l r -r cosq+ sinq = 9 h) Find the centre and radius of the circle ( ) a) - Sin d b) cos d c) e sin d on R æ sin ö d) e - ç d e) çè -cos ø d - -7 f) d - + g) d + + h) d i) + - d j) + cos cos + sin d cos + sin

8 6 a) Form the differential equation corresonding to the famil of circles of radius r given b ( ) ( ) - a + - b = r, where a and b are arameters b) Form the differential equation corresonding to the famil of curves ae be = + where a and b are arameters c) d + = d - d - = e) d d) ( ) a d - + = d - f) d d = - g) ( ) d d = + + h) d Sin - æ ö = + ç èdø d d = + j) ( ) d d i) tan ( ) l) o) d + d + = k) ( ) = m) ( + ) d= d n) ( ) ( + -9) d =- d + - ) d -- + = d + + d- + d= - d- d= q) ( ) d + + = + + d r) d + 6+ = d a) ( ) d d d d cos sin sec = b) ( ) e d + = + c) d + = d + + = e) ( + ) = f) ( ) d) ( ) d d d d + + = d g) d sin = cos d + h) æ ö d + ç - = çè ø d

9 7 MARKS QUESTIONS 8 a) Find the equation of the circle assing through (, ), (6, ) and having the centre on the line + - 6= b) Find the equation of the circle which asses through the vertices of the triangle formed b L= + + =, L = + - = and L = + - = c) Find the equation of the circle assing through (,) and making intercet 6 units on X-ais and intercet units on Y-ais d) Show that the following four oints (,)(, -),(, - 6 ),( 9,8) are concclic and find the equation of the circle on which the lie e) Find the equations of circles which touch - + = at (, ) and having radius f) Show that the tangent at (-,) of the circle = and also find its oint of contact g) Find all the common tangents of the circles = touches the circle = and + = h) Find the equation of the circle assing through (-,) and touching + - 7= at (, ) i) Find the equations of circles assing through (, - ), touching the lines + + = and - - = j) If q q are the angles of inclination of tangents through a oint P to the circle find the locus of P when cotq + cotq = k k) Show that the circles = and + = a then = touch each other Also find the oint of contact and common tangent at this oint of contact l) Find the common tangents of the given airs of circles = = and 9 a) Find the equation of the circle which asses through (, ) and cuts orthogonall each of the circles = and = b) Find the equation of the circle which intersects the circle and asses through the oint (,) and touches Y-ais c) Show that the common chord of the circles the diameter of the second circle and also find its length = orthogonall = and = is

10 d) Find the equation of the circle which cuts the following circles orthogonall =, =, = e) Find the limiting oints of coaial sstem of circles of which two members are S= = and a) Prove that the standard equation of arabola is S' = = = a b) Prove that the area of the triangle inscribed in the arabola 8a ( - )( - )( - ) sq units where c) Show that the common tangent to the arabola a + b + a + b = = a is,, are the ordinates of its vertices = a and = b is d) Prove that the area of the triangle formed b the tangents at (, ), (, ) and (, ) to the 6a arabola = a( a> ) is ( - )( - )( - ) sq units e) Find the eccentricit, coordinates of foci, Length of latus rectum and equations of directrices of the ellises = f) The tangent and normal to the ellise + = at a oint P( q ) on it meets the major ais in Q and R resectivel If <q< and QR= then show that - q= Cos æ ç ö çè ø g) Show that the oles of the tangents to the auiliar circle wrt the ellise S= lie on the curve + = a h) A chord PQ of an ellise S= subtends a right angle at the centre of the ellise Show that the oint of intersection of tangents at P and Q lies on another ellise + = + i) Show that the oles of normal chords of the ellise + = lie on the curve = ( a - b ) j) Prove that the oles of normal chords of the herbola - = lie on the curve = ( a + b )

11 a) If = a cos( log ) + bsin( log ), > then rove that + ( ) + ( ) n + n+ n + n + n = - =, b) If log Tan ( ) + n+ + n+ - n+ + n n+ n = >, then show that ( ) ( ) ( ) =, >, then show that + + m = and hence deduce that c) If cos( m log ) n+ + n+ n+ + m + n n = ( ) ( ) d) If Sin - = e, then show that ( ) = and hence deduce that e) If f) If ( ) + ( ) + ( ) - n - n+ n - n + n = = e log, >, then show that -( - ) + ( - ) = and hence deduce that ( ) ( ) n n n n n+ nn- = e - n+ + n+ + n+ n = =, then rove that ( ) g) If = ( - ) n, then rove that ( ) ( ) - n+ + n+ - n n+ n = h) If msin - = e, then rove that ( - ) n+ - ( n+ ) n+ - ( n + m ) n = a) Find d cos + sin + 6 b) Evaluate the following integrals i) + ii) ( 6+ ) 6- + d iii) - + d cos + sin iv) sin cos d cos + cos + c) If =, then show that n In cos d n- n- In cos sin+ In- n n = and hence find cos d d) Obtain reduction formula for =, n being a ositive integer, n³ and deduce the n In cot d value of cot d e) Obtain the reduction formula for the value of cos ec d In n = cos ec d, n being a ositive integer, n ³ and deduce

12 a) Show that d= log( + ) sin + cos b) Find sin d + sin c) Evaluate the following: i) sin + cos d 9+ 6sin ii) sin d + cos iii) ( + ) log d + iv) sin d + cos v) sin d cos + sin d) Find 7 6 sin cos d a) Show that the area enclosed between the curves = ( + ) and ( ) = - is 6 b) The circle arts + = 8 is divided into two arts b the arabola = Find the area of both the c) Dividing [,6] into equal arts, evaluate 6 d aroimatel b using (i) Traezoidal rule ii) Simson s rule d) Find the aroimate value of from equal arts d + using Simson s rule b dividing [, ] into e) Use Simson s rule to evaluate aroimate value of 7 d aroimatel b taking n= 6 and hence find the loge 7 to three laces of decimals Wwish ou all the bestw