Question
For the ellipse 3x2 + 4y2 – 6x + 8y – 5 = 0
- centre is (2, -1)
- eccentricity is
- foci are (3, 1) and (-1, 1)
- centre is (1, -1), e =
.
Hint:
Standard form of ellipse
![x squared over a squared plus y squared over b squared space equals space 1
a n d space e c c e n t r i c i t y left parenthesis e right parenthesis space equals space square root of 1 minus b squared over a squared end root](data:image/png;base64,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)
The correct answer is: centre is (1, -1), e =
.
Given :
![3 x squared space plus space 4 y squared space – space 6 x space plus space 8 y space – space 5 space equals space 0](data:image/png;base64,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)
Step 1 Convert in standard form
![rightwards double arrow 3 x squared space plus space 4 y squared space – space 6 x space plus space 8 y space – space 5 space equals space 0](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPsAAAATCAYAAABWZXtJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAABAhJREFUeNrtW39EXVEcP5JkMpJM9kckSfKMySQzI8kkicxkZmImM8/0X7I/ZmQmMzMymZkZSX8kz5gkkxnzTCaJzMxM/+yPJM8z7r5ffR6n69777rv3vHvOe50PH9679zrvnO+v8/1+z31CVA4cME/cJLaLyka1rcfCQjlqiJPErF2PhcXpQM6ux8Ki+nGFuGHXY2GhH1eJGeIRdqxd4jNik4KxW1DjthmwziHU3ZW0nh7iIvEAullBsDEZZ4jPiYeY8wfieQPm5QRQB84SXxK/gPO4VlasE0eJddK1C1BSHHQQV+EgutFA3I6p2KTXcwfO3SEZxzjxk+HOvkB8TKyFTc3CmE1wdpOwRhx2bUZruiZzEHMHzCiKVCqUxFFzIsZYKtcTBl3ErxXqGO5+Bjc189bZT2AM2Y8bnFHfiDLgVAzj5DR1x+P6BCK1G49wrwB2jE5DlNQnRUzHZye67nE9jR1K9XrCBqewSg+rk6RkfohMSnb2XxFkXs3OvkQc8Cmpl6IMOE38CWMPi3PE26jbr/k8k8VzBfDzL0LURzqUxGnkFrE1YKxbxCcedeeu1LdIus7bKbFcCKOTpGTOO9Z96Xsv8WkEmZvo7E4IhsFfV9ks2+t+nB93EC1rSlhEOuDZQeIcPvcnUGfEURLvePeKjMVBbdF17SFxRmPkz6FWXxbHjdM80vrxhHQSR+YpzJfJpxZ/iJcNkHnhpagDZGoPXBlIkvgXcC9WybOAhb4L8WwjcQQNlaCu70fc/1aGSKwqerLRbYYwYp7/d+l7M3GPWK+5vszCKWoRqPuw802WQSeqZN4KO2uRdq6LkGfKEJnXYk4zyKA6DXP2XNydfbbIzu5GgwjuoKYRgVIGN1Z4/u0hxzqSPs8VyWxUOFEx8O7jdVzFpyS/E9BJVJkv+/w+16LrimSu8vhsQJT2zoSqoLjvk8bXB6Xxqmv2MBGmD02EORGxc5iQ4ZWiFD7OagP3SgyM5UAmYA75BHTiKLYZr3umyFzH25BLKL28gk+kBt1UjJokhZTRDVbMCpopDagjmwx19lLGeiOOzzw5Bb0p9INTda9udadHxlUOnUSV+bbw/pNQF2p302TejUCTNPi9llce11+XewPlNGZMqg055fohjjumwlVXZVyGxC8CvK0CZ0+jt7ElzEAd9DIBvTAuYX79CejEiWHE27ChGsme2KHuapY5lxi90rwGMa9RTTpehwwKLx9NiwRew+ZO6SrSmRwmMeRhfMs+deR7n5TERPgZ8QjuDRs012ZE/0M0dDbEya62qTrhjSOLciOHeY8YIPMxZKssSz764tOAHo36bUSwy6F/MS/0nQ6cKrDBfbZisDK3qH5kkOJZWJlbVDG6UcZYWJkbg/+oXGUHN+jizwAAAbh0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjFEMjs8L21vPjxtbj4zPC9tbj48bXN1cD48bWk+eDwvbWk+PG1uPjI8L21uPjwvbXN1cD48bW8+JiN4QTA7PC9tbz48bW8+KzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1uPjQ8L21uPjxtc3VwPjxtaT55PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4mI3hBMDs8L21vPjxtbz4mI3gyMDEzOzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1uPjY8L21uPjxtaT54PC9taT48bW8+JiN4QTA7PC9tbz48bW8+KzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1uPjg8L21uPjxtaT55PC9taT48bW8+JiN4QTA7PC9tbz48bW8+JiN4MjAxMzs8L21vPjxtbz4mI3hBMDs8L21vPjxtbj41PC9tbj48bW8+JiN4QTA7PC9tbz48bW8+PTwvbW8+PG1vPiYjeEEwOzwvbW8+PG1uPjA8L21uPjwvbWF0aD7sy9ZPAAAAAElFTkSuQmCC)
![rightwards double arrow 3 x squared space – space 6 x italic space italic plus italic 3 italic space italic minus italic 3 plus space 4 y squared space space plus space 8 y italic space italic plus italic 4 italic space italic minus italic 4 space – space 5 space equals space 0
rightwards double arrow 3 x squared space – space 6 x italic space italic plus italic 3 italic space plus space 4 y to the power of italic 2 space space plus space 8 y italic space italic plus italic 4 italic space italic minus italic 12 italic space italic equals italic space italic 0
rightwards double arrow 3 left parenthesis x to the power of italic 2 space – space 2 x italic space italic plus italic 1 italic right parenthesis italic space italic plus italic space italic 4 italic left parenthesis y to the power of italic 2 space space plus space 2 y italic space italic plus italic 1 italic right parenthesis italic space italic space italic equals italic space italic 12
D i v i d i n g italic space b y italic space italic 12 italic space o n italic space b o t h italic space s i d e 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)
![fraction numerator italic rightwards double arrow italic left parenthesis x to the power of italic 2 italic space italic – italic space italic 2 x italic space italic plus italic 1 italic right parenthesis over denominator italic 4 end fraction italic space italic plus italic space fraction numerator italic left parenthesis y to the power of italic 2 italic space italic space italic plus italic space italic 2 y italic space italic plus italic 1 italic right parenthesis italic space italic space over denominator italic 3 end fraction italic equals italic space italic 1
rightwards double arrow italic left parenthesis x italic minus italic 1 italic right parenthesis to the power of italic 2 over italic 4 italic space italic plus italic space italic left parenthesis y italic space italic plus italic space italic 1 italic right parenthesis to the power of italic 2 over italic 3 italic space italic equals italic space italic 1
C o m p a r i n g italic space w i t h italic space s italic tan d a r d italic space e q u a t i o n
rightwards double arrow a to the power of italic 2 italic space italic equals italic space italic 4 italic space a n d italic space b to the power of italic 2 italic space italic equals italic 3
rightwards double arrow C o o r d i n a t e s italic space o f italic space c e n t r e italic space italic equals italic space italic left parenthesis italic 1 italic comma italic minus italic 1 italic right 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)
![E c c e n t r i c i t y left parenthesis e right parenthesis space equals space square root of 1 minus b squared over a squared end root
rightwards double arrow e space equals space square root of 1 minus fraction numerator 3 over denominator 4 space end fraction end root space equals space 1 half](data:image/png;base64,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)
Related Questions to study
If S’ and S are the foci of the ellipse
and P (x, y) be a point on it, then the value of SP + S’P is
If S’ and S are the foci of the ellipse
and P (x, y) be a point on it, then the value of SP + S’P is
The eccentricity of the curve represented by the equation x2 + 2y2 – 2x + 3y + 2 = 0 is
The eccentricity of the curve represented by the equation x2 + 2y2 – 2x + 3y + 2 = 0 is
The foci of the ellipse 25 (x + 1)2 + 9 (y + 2)2 = 225 are
The foci of the ellipse 25 (x + 1)2 + 9 (y + 2)2 = 225 are
The eccentricity of the ellipse 9x2 + 5y2 – 30y = 0 is
The eccentricity of the ellipse 9x2 + 5y2 – 30y = 0 is
The sum of all possible numbers greater than 10,000 formed by using {1,3,5,7,9} is
The sum of all possible numbers greater than 10,000 formed by using {1,3,5,7,9} is
Number of different permutations of the word 'BANANA' is
In this question, one might think that the total number of permutations for the word BANANA is 6!
This is wrong because in the word BANANA we have the letter A repeated thrice and the letter N repeated twice.
Number of different permutations of the word 'BANANA' is
In this question, one might think that the total number of permutations for the word BANANA is 6!
This is wrong because in the word BANANA we have the letter A repeated thrice and the letter N repeated twice.
The number of one - one functions that can be defined from
into
is
The number of one - one functions that can be defined from
into
is
Four dice are rolled then the number of possible out comes is
Four dice are rolled then the number of possible out comes is
The number of ways in which 5 different coloured flowers be strung in the form of a garland is :
The number of ways in which 5 different coloured flowers be strung in the form of a garland is :
Number of ways to rearrange the letters of the word CHEESE is
Number of ways to rearrange the letters of the word CHEESE is
The number of ways in which 17 billiard balls be arranged in a row if 7 of them are black, 6 are red, 4 are white is
We cannot arrange the billiards balls as follows:
There are 17 balls, so the total number of arrangements = 17!
This is incorrect because this method is applicable only when the balls are distinct, not identical. But, in our case, all the balls of the same color are identical.
The number of ways in which 17 billiard balls be arranged in a row if 7 of them are black, 6 are red, 4 are white is
We cannot arrange the billiards balls as follows:
There are 17 balls, so the total number of arrangements = 17!
This is incorrect because this method is applicable only when the balls are distinct, not identical. But, in our case, all the balls of the same color are identical.
Number of permutations that can be made using all the letters of the word MATRIX is
Number of permutations that can be made using all the letters of the word MATRIX is
The number of different signals that can be made by 5 flags from 8 flags of different colours is :
Each of the different arrangements which can be made by taking some or all of a number of things at a time is called permutation.
Thus we can say that the permutation concerns both the selection and the arrangement of the selected things in all possible ways.
The number of different signals that can be made by 5 flags from 8 flags of different colours is :
Each of the different arrangements which can be made by taking some or all of a number of things at a time is called permutation.
Thus we can say that the permutation concerns both the selection and the arrangement of the selected things in all possible ways.