S1 : If & are the equations of asymptotes of a hyperbola and hyperbola passes through the point then length of its latus rectum is .
S2 : Two concentric rectangular hyperbolas whose axes meet at an angle cut each other at an angle .
S3 : Distance between directrices of hyperbola is
S4 : If line joining the points & is tangent to the hyperbola then point of contact is
Equation of Tangent to Hyperbola -
- wherein
For the Hyperbola
and
Rectangular Hyperbola -
- wherein
S1 : Equation of hyperbola
It passes through then
Latus rectum
S2 : Let the equation to the rectangular hyperbola be
As the asymptotes of this are the axes of the other and vice-versa, hence the equation of the other hyperbola may be written as
Let (i) and (ii) meet at some point whose coordinates are
then the tangent at the point to equation on (i) is
and the tangent at the point to equation on (ii) is
So, the slopes of the tangents given by (iii) and (iv) are respectively and and their product is
Hence the tangents are right angle.
S3 : Hyperbola
equation of directrices
distance between directrices of hyperbola is
S4 : Let point on the parabola. then equation of tangent is
Equation of line
and
point of contact is
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