Ifthe normal at P to the rectangular hyperbola `x^2-y^2=4` meets the axes in G and g and C is the centre of the hyperbola, then

Question

Ifthe normal at P to the rectangular hyperbola `x^2-y^2=4` meets the axes in G and g and C is the centre of the hyperbola, then

Ifthe normal at P to the rectangular hyperbola `x^2-y^2=4` meets the axes in G and g and C is the centre of the hyperbola, then
A. `PG=PC`
B. `Pg=PC`
C. `PG=Pg`
D. `Gg=2PC`

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - A::B::C::D
Normal at point `P(2 sec theta, 2 tan theta)` is
`(2x)/(sec theta)+(2y)/(tan theta)=8`
It meets the axes at points `G(4 sec theta, 0) and g(0, 4 tan theta)`. Then,
`PG=sqrt(4 sec^(2)theta+4 tan^(2)theta)`
`Pg=sqrt(4 sec^(2)theta+4 tan^(2) theta)`
`PC=sqrt(4 sec^(2)theta+4 tan^(2)theta)`
`Gg=sqrt(16 sec^(2)theta+16 tan^(2)theta)`
`=2sqrt(4 sec^(2)theta+2 tan^(2)theta)=2PC`