If p and p- be the perpendiculars from the origin upon the straight lines whose equations are x - y cosec - - a and x cos- -ysin- - a cos 2-prove that 4p2-p-2- a2-

Perpendiculars from origin for the given lines are: p=|a√(sec^2θ+cosec^2θ)|=asin2θ2 ......(1) p^'=|acos2θ√(cos^2θ+sin^2θ)|=acos2θ ........(2) (2× p) ...

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Byju's Answer

Standard XII

Mathematics

Derivative of Standard Functions

If p and ...

Question

If $p$ and $p^{'}$ be the perpendiculars from the origin upon the straight lines whose equations are $x sec θ + y c o s e c θ = a$ and
$x cos θ - y sin θ = a cos 2 θ$ ,
prove that $4 p^{2} + p^{' 2} = a^{2}$ .

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Solution

Perpendiculars from origin for the given lines are:
$p = ∣ ∣ ∣ \frac{a}{\sqrt{s e c^{2} θ + c o s e c^{2} θ}} ∣ ∣ ∣ = \frac{a s i n 2 θ}{2}$ ......(1)
$p^{^{'}} = ∣ ∣ ∣ \frac{a c o s 2 θ}{\sqrt{c o s^{2} θ + s i n^{2} θ}} ∣ ∣ ∣ = a c o s 2 θ$ ........(2)
$(2 \times p)^{2} + (p^{'})^{2} = (a s i n 2 θ)^{2} + (a c o s 2 θ)^{2}$
$\Rightarrow 4 p^{2} + (p^{'})^{2} = a^{2}$

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Similar questions

Q. If

p

and

q

are respectively the perpendiculars from the origin upon the straight lines, whose equations are

x sec θ + y c o s e c θ = a

and

x cos θ - y sin θ = a cos 2 θ

, then

4 p^{2} + q^{2}

is equal to

Q. If

P

and

P^{'}

are the perpendiculars from the origin, upon the straight lines

x sec θ + y csc θ = a

and

x cos θ - y sin θ = a cos 2 θ,

then the value of

4 P^{2} + {P^{'}}^{2}

Q. If

p

and

p^{'}

be the lengths of perpendiculars from origin
to the lines

x sec θ - y cos θ = a

and

x cos θ - y sin θ = a cos 2 θ

respectively, then prove
that

4 p^{2} + p^{' 2} = a^{a}

Q. If

p

and

p^{'}

are the perpendiculars from the origin upon the lines

x sec θ + y csc θ = a

and

x cos θ - y sin θ = a cos 2 θ

respectively then

Q. If

p

and

q

are the perpendicular distances from origin to the straight lines

x sec θ - y c o s e c θ = a

and

x cos θ + y sin θ = a cos 2 θ

then which of the following is correct?

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If p and p′ be the perpendiculars from the origin upon the straight lines whose equations are xsecθ+ycosecθ=a andxcosθ−ysinθ=acos2θ,prove that 4p2+p′2=a2.

Perpendiculars from origin for the given lines are:p=∣∣∣a√sec2θ+cosec2θ∣∣∣=asin2θ2......(1)p′=∣∣∣acos2θ√cos2θ+sin2θ∣∣∣=acos2θ........(2)(2×p)2+(p′)2=(asin2θ)2+(acos2θ)2⇒4p2+(p′)2=a2

If $p$ and $p^{'}$ be the perpendiculars from the origin upon the straight lines whose equations are $x sec θ + y c o s e c θ = a$ and
$x cos θ - y sin θ = a cos 2 θ$ ,
prove that $4 p^{2} + p^{' 2} = a^{2}$ .