Question

Find the distance between PQ, P(a sin x, a cos x) and Q (a cos x, - a sin x)​

Answer 1

Given

⇒P(aSinx , aCosx) , Q( aCosx , -aSinx )

To find

⇒Distance between PQ

Formula

⇒D = √{(x₂ - x₁)² + (y₂ - y₁)²}

We have

⇒x₁ = aSinx , y₁ = aCosx , x₂ = aCosx and y₂= -aSinx

Put the value on formula

⇒PQ=√{(aCosx - aSinx)² + (-aSinx - aCosx)²}

⇒PQ = √{ (a²Cos²x + a²Sin²x - 2a²CosxSinx) + (a²Sin²x + a²Cos²x + 2a²SinxCosx)}

⇒PQ = √{ a²(Cos²x +Sin²x - 2CosxSinx) + a²(Sin²x + Cos²x + 2SinxCosx)}

⇒PQ = a√{(Cos²x +Sin²x - 2CosxSinx) + (Sin²x + Cos²x + 2SinxCosx)}

We know that

⇒Sin²Ф + Cos²Ф = 1

⇒2SinФCosФ = Sin2Ф

We get

⇒PQ=a√{(1 - Sin2x) + ( 1 + Sin2x)}

⇒PQ = a√{1 - Sin2x + 1 + Sin2x}

⇒PQ = a√{2}

⇒PQ = a√2

Distance Between PQ = a√2

Answer 2

$\large\underline\blue{\bold{Given \:  Question :-  }}$

• Find the distance between PQ, P(a sin x, a cos x) and Q (a cos x, - a sin x)

Answer

$\begin{gathered}\begin{gathered}\bf \: Given \: -   \begin{cases} &\sf{coordinates \: of \: P(a sin x, a cos x)} \\ &\sf{coordinates \: of \: Q (a cos x, - a sin x)} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: Find  - \begin{cases} &\sf{distance \: between \: P \: and \: Q}  \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\Large{\bold{\green{\underline{Formula \: Used \::}}}} \end{gathered}$

Distance Formula

Let us consider a line segment joining the points A and B, then distance between A and B is given by

$\boxed{ \pink{\rm\implies \:AB = \sqrt{ {(x_2-x_1)}^{2} + {(y_2-y_1)}^{2} }}}$

where,

$\rm \: coordinates \: of \: A \: and \: B \: are \: (x_1,y_1) \: and \: (x_2,y_2).$

$\large\underline\purple{\bold{Solution :- }}$

Given,

Coordinates of P is (a sin x, a cos x)

and

Coordinates of Q is (a cos x, - a sin x)

So, Distance between P and Q using distance formula is

$\rm : \implies \:PQ \: = \sqrt{ {(acosx - asinx)}^{2} + {( - asinx - acosx}^{2} }$

$\rm : \implies \:PQ \: = \sqrt{ {(acosx - asinx)}^{2} + {( asinx + acosx}^{2} }$

$\rm : \implies \:PQ = \sqrt{2( {a}^{2} {cos}^{2} x + {a}^{2} {sin}^{2} x} )$

$\boxed {\because \: \pink{ \rm{(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2}) }}$

$\rm : \implies \:PQ = \sqrt{2{a}^{2} ({cos}^{2} x + {sin}^{2} x} )$

$\rm : \implies \:PQ = \sqrt{ {2a}^{2} }$

$\rm : \implies \:PQ \: = a \sqrt{2}$

Explore more :-

$\underline{\bigstar\:\textsf{Section Formula\; :}}$

• Section Formula is used to find the co ordinates of the point(Q) Which divides the line segment joining the points (B) and (C) internally.

${\underline{\boxed{\rm{\quad \Big(x, y \Big) = \Bigg(\dfrac{mx_2 + nx_1}{m + n} \dfrac{my_2 + ny_1}{m + n}\Bigg) \quad}}}}$

$\underline{\bigstar\:\textsf{Mid Point Formula\; :}}$

• Mid Point formula is used to find the Mid points on any line.

${\underline{\boxed{\rm{\quad \dfrac{x_1 + x_2}{2} \; or\; \dfrac{y_1 + y_2}{2} \quad}}}}$