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Conic Sections is an extremely important topic of IIT JEE Mathematics. The conics like circle, parabola, ellipse and hyperbola are all interrelated and therefore it is crucial to know their distinguishing features as well as similarities in order to attempt the questions in various competitive exams like the JEE. Hence, students are advised to study these topics attentively in order to remain competitive in the JEE.
In this section, we shall discuss the similarities and differences between ellipse and hyperbola in detail.
The standard equation of ellipse is x^{2}/a^{2} + y^{2}/b^{2} = 1, where a > b and b^{2 }= a^{2}(1 – e^{2}), while that of a hyperbola is x^{2}/a^{2} – y^{2}/b^{2} = 1, where b^{2 }= a^{2}(e^{2} -1). Hence, the equations are closely related, the only difference being of a negative sign.
Both ellipses as well as hyperbolas have vertices, foci, and a center. In standard form, both the curves have (0, 0) as the centre. As discussed above, in an ellipse, ‘a’ is always greater than b. if ‘a’ is greater than ‘b’ and ‘a’ lies below the term of x^{2} then the major axis is horizontal and similarly, if it lies under the y^{2} term, then the axis is vertical.
The situation is quite different in case of a hyperbola. In hyperbola, ‘a’ may be greater than, equal to or less than ‘b’. The order of the terms x^{2} and y^{2} decide whether the transverse axis would be horizontal or vertical. If x^{2} comes first then the transverse axis would be horizontal. Conversely, if the y^{2} term is listed first, the transverse axis would be vertical.
Going into the intricacies, we see that the focal distance in hyperbola is greater than the distance of the vertex. The situation is quite different in case of ellipse; the distance from the centre to vertex is greater than the focal distance.
Another striking difference between the both is that hyperbolas have asymptotes which ellipses don’t have.
We now proceed towards the various formulae and compare them for both:
Tangent at (x_{1}, y_{1}):
In case of hyperbola, the equation is xx_{1}/a^{2} – yy_{1}/b^{2} = 1 i.e. T = 0. The equation of tangent in case of ellipse is xx_{1}/a^{2} + yy_{1}/b^{2} = 1. Hence, again the only difference is of negative b^{2}.
Equation of tangent in slope form:
?In terms of ‘m’, the equation of tangent to a hyperbola is y = mx + √(a^{2}m^{2 }– b^{2}), while in case of ellipse, the equation is y = mx + √(a^{2}m^{2 }+ b^{2}).
Equation of the normal at (x_{1}, y_{1}):
In case of ellipse it is a^{2}x/x_{1}^{ }–^{ }b^{2}y/y_{1} = a^{2}-b^{2},^{ }and for hyperbola, the equation becomes a^{2}x/x_{1}^{ }+^{ }b^{2}y/y_{1} = a^{2}-b^{2}.
Equation of pair of tangents drawn from point (x_{1}, y_{1}):
Pair of tangents to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 is given by SS_{1} = T^{2},^{ }where S = x^{2}/a^{2} – y^{2}/b^{2} = 1, S_{1} = x_{1}^{2}/a^{2} – y_{1}^{2}/b^{2} = 1 and T = xx_{1}/a^{2} – yy_{1}/b^{2} = 1
Chord of Contact:
The chord of contact of tangents from (x_{1}, y_{1}) to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 is given by T = 0 i.e. xx_{1}/a^{2} – yy_{1}/b^{2} = 1 and by changing b^{2} to -b^{2}, we get the equation of chord of contact of tanegnts to an ellipse as xx_{1}/a^{2} + yy_{1}/b^{2} = 1.
Polar:
The Polar of the Pole (x_{1}, y_{1}) to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 is given by T = 0 i.e. xx_{1}/a^{2} – yy_{1}/b^{2} = 1.
Chord whose middle point is given:
?The equation of Chord of hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 whose middle point is (x_{1}, y_{1}) is given by T = S_{1} i.e. x_{1}^{2}/a^{2} – y_{1}^{2}/b^{2} = xx_{1}/a^{2} – yy_{1}/b^{2} = 1.
You may refer the following video for more on hyperbola and ellipse
Question:
Can we represent a hyperbola mathematically in form of one parameter and is there any geometrical significance of that parameter like eccentric angle in case of ellipse?
Answer:
The simple answer to this question is yes. Before that, let us understand the concept of the auxiliary circle of a hyperbola. The circle described on the transverse axis of hyperbola as its diameter is called its auxiliary circle.
We know that the line AA’ joining the vertices A(a, 0) and A’(–a, 0) of the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 is called the transverse axis.
Hence, the equation of the auxiliary circle, described on AA’ as diameter, is
(x – a) [x – (–a)] + (y – 0)] + (y – 0)(y – 0) = 0
or x^{2} + y^{2} = a^{2}
Now let us draw the foot N of any ordinate NP of the hyperbola draw a tangent NU to this circle, and join CU. Then
CU = CN cos NCU
i.e. x = CN = a sec NCU
The angle NCU is therefore the angle ∅.
Also NU = CU tan ∅ = a tan ∅
So that NP : NU = b : a
So the ordinate of the hyperbola is therefore in a constant ratio to the length of the tangent drawn from its foot to the auxiliary circle.
When it is desirable to express the co-ordinates of any point of the curve in terms of one parameter then we use x = a sec ∅, and y = b tan ∅.
This angle ∅ is not so important an angle for the hyperbolas the eccentric angle is for the ellipse.
A tangent to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 cuts the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1 in points L and R. Find the locus of the mid-point of LR.
Solution:
Let A(x_{1}, y_{1}) be the mid-point of the chord LR of the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1.
The equation of LR is xx_{1}/a^{2} + yy_{1}/b^{2} = x_{1}^{2}/a^{2} + y_{1}^{2}/b^{2}
This gives y = -b^{2}xx_{1}/_{ }a^{2}y_{1 }+_{ }b^{2}/y_{1}(x_{1}^{2}/a^{2} + y_{1}^{2}/b^{2})
This is a tangent to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 if
b^{4}/y_{1}^{2} (x_{1}^{2}/a^{2} + y_{1}^{2}/b^{2})^{2} = a^{2}b^{4}x_{1}^{2}/ a^{4}y_{1}^{2} – b^{2}
Hence, we have (x_{1}^{2}/a^{2} + y_{1}^{2}/b^{2})^{2} = x_{1}^{2}/a^{2} – y_{1}^{2}/b^{2}
Hence, the locus of (x_{1}, y_{1}) is (x^{2}/a^{2} + y^{2}/b^{2})^{2} = x^{2}/a^{2} – y^{2}/b^{2}.
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