a) 9x²+ 16y²= 144. (0,0),8,6,y=0, x=0, (±4,0), √7/4, (±√7,0), x=±16/√7, 9/2
b) x²+ 4y²= 1. (0,0),2,1,y=0, x=0, (±1,0),√3/2, (±√3/2,0), √3x=±2, 1/2
c) 9x²+ 4y²= 36. (0,0),6,4,x=0, y=0, (0,±3), √5/3, (0,±√5), y=±9/√5, 8/3
d) 5x²+ 4y²= 1. (0,0),1,2/√5,x=0, y=0, (0,±1/2), 1/√5, (0,± 1/2√5), 2y=±√5, 4/5
2) Find (i) the co-ordinate of centre (ii) lengths of axis (iii) equations of axis (iv) co-ordinates of vertices (v) eccentricity (vi) coordinates of focus (vii) equation of directrices (viii) length of latus rectum for each of the following ellipse :
a) 4x²+ 3y² - 16x + 12y + 16 =0. (2,-2),4,2√3, x-2=0, y+2=0, (2,0) and (2,-4), 1/2, (2,-1) and (2,-3), y -2=0 and y+ 6=0, 3
b) 4x²+ 9y² - 24x =0. (3,0),6,4, y=0, x -3=0, (0,0) and (6,0), √4/3, (3-√5,0) and (3+√5,0), x =3 - 9/√5 and x =3+ 9/√5, 8/3
c) 9x²+ 16y² - 54x + 64y + 1 =0. (3,-2),8,6, y+2=0, x=3, (-1,-2) and (7,-2), √7/4, (3±√7,-2), x=3±16/√7, 9/2
d) 9x²+ 5y² + 30y =0. (0,-3),6,2√5, y+3=0, x=0, (0,0) and (0,-6), 2/3, (0,-1) and (0,-5), 2y -3=0 and 2y+ 15 =0, 10/3
3) The ellipse x²/169 + y²/25 = 1 has the same eccentricity as the ellipse x²/a² + y²/b² = 1. Find the ratio a/b. 13/5
4) Taking x-axis as major axis and y-axis as minor axis, find the equation of the ellipse whose
a) length of the minor axis and latus rectum are 4 and 2 units respectively. x²+ 4y²= 16
b) length of latus rectum is 6 units and length of major axis is 10 units. 3x²+ 5y²=75
c) length of major axis is 4 units and eccentricity 3/4. 7x²+ 16y²= 28
d) distance between directrices is 16/√7 units and eccentricity is √7/4. 9x²+ 16y²=36
e) distance between the foci is 8 units and distance between the directrices is 18 units. x²/36 + y²/20 = 1
f) distance between the foci is equals to minor axis and length of the latus rectum is 16 units . x²+ 2y²=256
g) Sum of the squares of the lengths of major and minor axis is 20 and eccentricity is 1/√3. 2x²+ 3y²= 6
h) eccentricity is 1/2 and distance between the foci is 4 units . 3x²+ 4y²= 48
i) distance between the foci is 4√3 units and length of the minor axis is 4 units. Find also the eccentricity . x²+ 4y²= 16, √3/2
j) distance between the foci is 10 units and length of the latus rectum is 15 units . 3x²+ 4y²= 300
k) latus rectum is 5 and eccentricity is 2/3. 20x²+ 36y²= 405
5) Find the equation of the ellipse whose vertices are at the point (0,±10) and eccentricity is 4/5. 25x²+ 9y²=900
6) Find the equation of the ellipse whose vertices are at the points (±1/4,0) and the equations of directrices are x= ± 5/12. 16x²+ 25y²= 1
7) If the co-ordinates of the focus of an ellipse are (0,±8) and its eccentricity is 4/5, find its equation. 25x²+ 9y²= 900
8) Find the equation of the ellipse whose co-ordinates of foci are (±5,0) and (±4,0) respectively. 9x²+ 25y²= 225
9) The co-ordinate of the foci of an ellipse are (1,0) and (-1,0) and the length of the minor axis 2 units. Find the equation of the ellipse. x²+ 2y²= 2
10) The co-ordinates of foci of an ellipse are (0,2) and (0,-2). If its latus rectum is 6 units, find its equation. 4x²+ 3y²= 48
11) If the co-ordinates of the foci of an ellipse which passes through (4,1) are (±3,0), find the equation of the ellipse . x²+ 2y²= 18
12) If the co-ordinates of the foci of an ellipse are (±2,0) and the equations of the directrices are 2x= ±9, find the ellipse. 5x²+ 9y²= 45
13) The major axis of an ellipse is parallel to x-axis, the co-ordinates of its centre are (-2,3) eccentricity is 1/√3 and the length of the lactus rectum is 4 units . Find the equation of the ellipse . 2x²+ 3y² + 8x - 18y +17=0
14) The major axis of an ellipse is along y-axis, centre is at (0,2), length of the major axis is 6 units and eccentricity is 1/2 0; find the equation of the ellipse. 4x²+ 3y² - 12y - 15=0
15) the major axis of an ellipse is along y- 2 =9, minor axis is along x- 3 =0, eccentricity is 1/2 and length of the latus rectum is 3√3/2; find the equation of the ellipse . 3x²+ 4y² -18x - 16y +34=0
16) The ellipse x²/a² + y²/b² =1 passes through the point (-3,2) and its eccentricity is √3/5; find the length of the latus rectum. 4√19/5
17) Find the equation of the ellipse whose axes are co-ordinate axis and which passes through the points (-3,1) and (2,-2). Find also eccentricity of the ellipse. 3x²+ 5y² = 32 or 5x²+ 3y²=32, √2/5
18) The length of the latus rectum of an ellipse is 8 units and that of the major exis, which lies along the the x-axis, is 18 units . Find the equation in the standard form.
Determine the co-ordinates of the foci and the equations of it directrices . 4x²+ 9y² = 324, (±3√5,0), √5 x = ±27
19) The co-ordinates of the vertices of an ellipse are (-5,-1) and (-1,-1) and eccentricity is 1/3; find the equation of the ellipse . 8(x+ 3)² + 9(y +1)²= 32
20) The co-ordinates of the foci of an ellipse are (2,-2) and (2,6) and its eccentricity is 1/2; find the equation of the ellipse. 4(x -2)²+ 3(y -2)²= 192
21) If the foci of an ellipse be (2,3) and (-2,3) and semi-minor axis be √5, find the equation of the ellipse. 5x²+ 9(y -3)² =45
22) The co-ordinates of the centre of an ellipse are (-2,1) and the co-ordinates of a focus are (-1,1); if the length of the major axis of the ellipse is 2√3 units , find its equation. 2(x +2)²+ 3(y -1)² =6
23) Show that the point (2,5/3) lies on the ellipse 5x²+ 9y² =45. Show further that the sum of the distance of this point from the foci is equal to the length of the major axis. √3/2
24) respectivali of the leaves is 12 find the equation the coordinate of the centre of an ellipse 21 and a coordinate of the focus are 11 of the length of the major Axis 23 units find its equation so the point is 253 lies on the lips so for that the sum of the distance of the point from the focus equals to the length of the major excessity of the ellipso that the sum of the distance of any point and the lips on the pokey is equals to 8 units find the equation of Delhi coordinate of the focus has 23 equation of the directories is nsentitive 12 the coordinate of the focus of an ellipse is 03 the equation of the corresponding directors and it is a cities 12 find the equation of an leaf with the centre city of an ellipse 13 focuses at the point 21 and the point of intersection of major X is directorate is 23 then find the coordinate of the centre of Delhi the coordinate of the focus and its corresponding but its upon A list at 31:24 respectively the central city of the ellipse is 23 then find the coordinator 7676 12 find the equation of find the equation of the auxiliary circle a piece of the following ellipse prove that the point lies on the ellipse if you came the variable quantity so that the locus of the point of intersection of the state lines is an ellipse find the co-ordinate of the point of the ellipse whose eccentricity angle is 30 find the centric angle of the point on the ellipse whose the distance from the centre is 3 units find the century candles of the experimentals of the late to select of the lips so that formali the distance between focus is Q and the distance between two focus 2P find the length of the semi Axis so that the double ordinate of the auxiliary circle up passing through the focus is equal to the minor x aadhe centric angle of two points PQ respectively on the ellipse than so that the equation of the chord will be if the eccentric angle of the extremities of a focal cord of the ellipse so that is an eccentricity to e so that the length of the focal drawn through end of the manufacturing series if the segment PQ find the locus of the point of action nearest to p