NIMCET 2008 Question Paper with solutions

Question : 6 of 120

Marks: +1, -0

A line L has intercepts 'a' and 'b' on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line has intercepts 'p' and 'q' which of the following statements is true?

$a^{2}+b^{2}=p^{2}+q^{2}$
$\;{1}/{a^{2}}+\;{1}/{b^{2}}=\;{1}/{p^{2}}+\;{1}/{q^{2}}$
$a^{2}+p^{2}=b^{2}+q^{2}$
$\;{1}/{a^{2}}+\;{1}/{p^{2}}=\;{1}/{b^{2}}+\;{1}/{q^{2}}$

Validate

Solution:

The line

L

will be

\;{x}/{a}+\;{y}/{b}=1

x y-

coordinate system. When the axes are rotated by an angle '

θ

in anti-clockwise direction.

x^{'}=x \cos θ+y \sin θ

y^{'}=-x \sin θ+y \cos θ

.....(i)

⇒ x=x^{'} \cos θ-y^{'} \sin θ

y=x^{'} \sin θ+y^{'} \cos θ

⇒ \Line \is {x^{'} \cos θ-y^{'} \sin θ}/{a}+ {x^{'} \sin θ+y^{'} \cos θ}/{b}-1

⇒ x^{'}[ {\cos θ}/{a}+ {\sin θ}/{b}]+y+ {\cos θ}/{b}- {\sin θ}/{a}]=1

⇒ Intercept p and q are

p=\;{a b}/{b \cos θ+a \sin θ} ; q=\;{a b}/{a \cos θ-b \sin θ}

⇒ {1}/{p^{2}}+\;{1}/{q^{2}}= {[\table a^{2} \sin ^{2} θ+b^{2} \cos ^{2} θ ; +a^{2} \cos ^{2} 0+b^{2} \sin ^{2} θ]}/{a^{2} b^{2}}= {a^{2}+b^{2}}/{d b^{2}}

=\;{1}/{a^{2}}+\;{1}/{b^{2}}

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