Understand the Equation of Hyperbola. Click on the Link to Watch the VIDEO explanation:
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Equation of the Hyperbola
Equation of the hyperbola in the form of x square by a square minus y square by b square is equal to 1
Hyperbola is the locus of a point which moves such that the ratio of its distance from the focus to its distance from the directrix is greater than 1.
S is the focus and l is the directrix. Draw SZ perpendicular to l.
On SZ mark the points A and A dash such that
SA by AZ is equal to SA dash by A dash Z equal to e by 1
Therefore, SA is equal to eAZ let this be the equation 1
SA dash is equal to eA dash Z let this be the equation 2
Bisects AA dash at C.
Take C as the origin, CS produced as X - axis and CY perpendicular to CS as Y - axis.
Let P (x, y) be any point on the hyperbola. Join PS.
Draw PM is perpendicular to the directrix, PN is perpendicular to x - axis. Take CA equal to CA dash equal to a.
Adding equations 1 and 2
SA plus SA dash is equal to e into AZ plus A dash Z
i.e., CA minus CA plus CS plus CA is equal to e into AA dash
i.e., 2cs is equal to e into 2a since AA dash is equal to 2a
Therefore, CS is equal to ae. Therefore, Coordinates of S are ae,0
Subtracting 1 from equation 2, then SA dash minus SA is equal to e into A dash Z minus AZ
i.e., AA dash is equal to e into CZ plus CA dash minus CA minus CZ
i.e., 2a is equal to e. 2cz. Therefore, CZ is equal to a by e.
Therefore Coordinates of z are a by e , 0
By distance formula, PS is equal to root x minus ae whole square plus y minus 0 whole square
From the fig. , PM is equal to NZ is equal to CN minus CZ is equal to x minus a by e Since CN is equal to x
Therefore, PM is equal to x minus a by e
Since P is a point on the hyperbola,
PS by PM is equal to e i.e., Ps is equal to e.PM
Therefore root of x minus ae the whole square plus y minus 0 the whole square is equal to e into x minus a by e
i.e., root of x minus ae the whole square plus y square is equal to ex minus a
Squaring both sides and simplifying
x square into 1 minus e square plus y square is equal to a square 1 minus e square
Multiply by minus 1
minus x square into 1 minus e square minus y square is equal to minus a square into 1 minus e square
i.e., x square into e square minus 1 minus y square is equal to a square into e square minus 1
x square by a square minus y square by a square into e square minus 1 is equal to 1
Since, e is greater 1, a square into e square minus 1 is positive.
Therefore, Take a square into e square minus 1 is equal to b square
Therefore, x square by a square minus y square by b square is equal to 1.
This is the equation of the hyperbola.
Watch Video
Equation of the Hyperbola
Equation of the hyperbola in the form of x square by a square minus y square by b square is equal to 1
Hyperbola is the locus of a point which moves such that the ratio of its distance from the focus to its distance from the directrix is greater than 1.
S is the focus and l is the directrix. Draw SZ perpendicular to l.
On SZ mark the points A and A dash such that
SA by AZ is equal to SA dash by A dash Z equal to e by 1
Therefore, SA is equal to eAZ let this be the equation 1
SA dash is equal to eA dash Z let this be the equation 2
Bisects AA dash at C.
Take C as the origin, CS produced as X - axis and CY perpendicular to CS as Y - axis.
Let P (x, y) be any point on the hyperbola. Join PS.
Draw PM is perpendicular to the directrix, PN is perpendicular to x - axis. Take CA equal to CA dash equal to a.
Adding equations 1 and 2
SA plus SA dash is equal to e into AZ plus A dash Z
i.e., CA minus CA plus CS plus CA is equal to e into AA dash
i.e., 2cs is equal to e into 2a since AA dash is equal to 2a
Therefore, CS is equal to ae. Therefore, Coordinates of S are ae,0
Subtracting 1 from equation 2, then SA dash minus SA is equal to e into A dash Z minus AZ
i.e., AA dash is equal to e into CZ plus CA dash minus CA minus CZ
i.e., 2a is equal to e. 2cz. Therefore, CZ is equal to a by e.
Therefore Coordinates of z are a by e , 0
By distance formula, PS is equal to root x minus ae whole square plus y minus 0 whole square
From the fig. , PM is equal to NZ is equal to CN minus CZ is equal to x minus a by e Since CN is equal to x
Therefore, PM is equal to x minus a by e
Since P is a point on the hyperbola,
PS by PM is equal to e i.e., Ps is equal to e.PM
Therefore root of x minus ae the whole square plus y minus 0 the whole square is equal to e into x minus a by e
i.e., root of x minus ae the whole square plus y square is equal to ex minus a
Squaring both sides and simplifying
x square into 1 minus e square plus y square is equal to a square 1 minus e square
Multiply by minus 1
minus x square into 1 minus e square minus y square is equal to minus a square into 1 minus e square
i.e., x square into e square minus 1 minus y square is equal to a square into e square minus 1
x square by a square minus y square by a square into e square minus 1 is equal to 1
Since, e is greater 1, a square into e square minus 1 is positive.
Therefore, Take a square into e square minus 1 is equal to b square
Therefore, x square by a square minus y square by b square is equal to 1.
This is the equation of the hyperbola.
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