If x=5+2 sec θ and y=5+2 tan θ , then (x-5)2-(y-5)2 is equal to

Question

If x=5+2 sec θ and y=5+2 tan θ , then (x-5)2-(y-5)2 is equal to (A) 3 (B) 1 (C) 0 (D) 4

Tardigrade Admin · Accepted Answer

Given, x=5+2 sec θ ⇒ x-5=2 sec θ ..(i) and y=5+2 tan θ ⇒ y-5=2 tan θ ....(ii) From Eqs. (i) and (ii), we get (x-5)2-(y-5)2 = (2 sec θ)2-(2 tan θ)2 = 4 sec 2 θ-4 tan 2 θ=4( sec 2 θ- tan 2 θ)=4

Q. If $x = 5 + 2$ sec $θ$ and $y = 5 + 2 tan θ,$ then $(x - 5)^{2} - (y - 5)^{2}$ is equal to

3

1

0

4

2

Given, $x = 5 + 2 sec θ$
$\Rightarrow x - 5 = 2 sec θ .. (i)$
and $y = 5 + 2 tan θ$
$\Rightarrow y - 5 = 2 tan θ .... (ii)$
From Eqs. (i) and (ii), we get
$(x - 5)^{2} - (y - 5)^{2}$
$= (2 sec θ)^{2} - (2 tan θ)^{2}$
$= 4 sec^{2} θ - 4 tan^{2} θ = 4 (sec^{2} θ - tan^{2} θ) = 4$

Q. If x=5+2 sec θ and y=5+2tanθ, then (x−5)2−(y−5)2 is equal to

3

1

0

4

2

Given, x=5+2secθ ⇒x−5=2secθ..(i) and y=5+2tanθ ⇒y−5=2tanθ....(ii) From Eqs. (i) and (ii), we get (x−5)2−(y−5)2 =(2secθ)2−(2tanθ)2 =4sec2θ−4tan2θ=4(sec2θ−tan2θ)=4

Q. If $x = 5 + 2$ sec $θ$ and $y = 5 + 2 tan θ,$ then $(x - 5)^{2} - (y - 5)^{2}$ is equal to

Given, $x = 5 + 2 sec θ$
$\Rightarrow x - 5 = 2 sec θ .. (i)$
and $y = 5 + 2 tan θ$
$\Rightarrow y - 5 = 2 tan θ .... (ii)$
From Eqs. (i) and (ii), we get
$(x - 5)^{2} - (y - 5)^{2}$
$= (2 sec θ)^{2} - (2 tan θ)^{2}$
$= 4 sec^{2} θ - 4 tan^{2} θ = 4 (sec^{2} θ - tan^{2} θ) = 4$