Please explain the answer

Asked by yasharthshankar | 10th Jun, 2020, 08:36: AM

Expert Answer:

At a point A(1, 1) on ellipse, equation of tangent is y=x. If one of the foci of ellipse if (0, -2) and the coordinates of centre are (alpha, beta) then the value of alpha+beta is:
(Given length of major axis of ellipse is 4sqrt10 units)
Solution:-
E q u a t i o n space o f space t h e space e l l i p s e space i s colon space open parentheses x minus alpha close parentheses squared over a squared plus open parentheses y minus beta close parentheses squared over b squared equals 1
O n e space o f space t h e space f o c i space o f space t h e space e l l i p s e space i s space g i v e n space a s space left parenthesis 0 comma space minus 2 right parenthesis
F o c i space equals space open parentheses alpha plus-or-minus a e comma space beta close parentheses space equals space left parenthesis 0 comma space minus 2 right parenthesis
rightwards double arrow space beta equals negative 2 space a n d space alpha plus-or-minus a e equals 0
rightwards double arrow a e equals plus-or-minus alpha space space o r space space alpha equals plus-or-minus a e
L e n g t h space o f space m a j o r space a x i s space i s space g i v e n space a s space 4 square root of 10 space u n i t s
2 a space equals space 4 square root of 10 space space rightwards double arrow space a equals 2 square root of 10
E q u a t i o n space o f space tan g e n t space i s space g i v e n space b y colon
y minus beta equals m left parenthesis x minus alpha right parenthesis plus-or-minus square root of a squared m squared minus b squared end root
y equals m x plus beta minus m alpha plus-or-minus square root of a squared m squared minus b squared end root subscript blank
E q u a t i o n space o f space tan g e n t space i s space g i v e n space a s space y equals x
rightwards double arrow m equals 1 space space a n d space space beta minus m alpha plus-or-minus square root of a squared m squared minus b squared end root equals 0
rightwards double arrow negative 2 minus alpha plus-or-minus square root of a squared minus b squared end root equals 0
rightwards double arrow plus-or-minus square root of a squared minus b squared end root equals alpha plus 2
rightwards double arrow plus-or-minus square root of a squared e squared end root equals alpha plus 2 space space space.... space S i n c e space e squared equals 1 minus b squared over a squared
rightwards double arrow plus-or-minus a e equals alpha plus 2
rightwards double arrow plus-or-minus alpha equals alpha plus 2
rightwards double arrow alpha equals negative 1
H e n c e comma space alpha plus beta equals negative 1 minus 2 equals negative 3

Answered by Renu Varma | 12th Jun, 2020, 01:54: PM

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