Name: The Standard Equation of a Circle
Uploaded: 2019-07-19
Duration: 2 min 23 s
Description: We all understand what a circle is. But do you know how a circle is defined exactly? What is the standard equation of a circle? Watch this video to know the answer to these questions.

Question

Prove that

x \cos θ + y \sin θ = a

and

x \sin θ - y \cos θ = b

are the parametric equations of a circle for all

θ

satisfying

0 \leq θ < 2 π

Accepted Answer

Given, $x \cos θ + y \sin θ = a & x \sin θ - y \cos θ = b$

$\Rightarrow {(x \cos θ + y \sin θ)}^{2} = a^{2} & {(x \sin θ - y \cos θ)}^{2} = b^{2}$

$\Rightarrow x^{2} \cos^{2} θ + y^{2} \sin^{2} θ + x y . 2 \sin θ \cos θ = a^{2} & x^{2} \sin^{2} θ + y^{2} \cos^{2} θ - x y . 2 \sin θ \cos θ = b^{2}$

$\Rightarrow (x^{2} \cos^{2} θ + y^{2} \sin^{2} θ + x y . \sin 2 θ) + (x^{2} \sin^{2} θ + y^{2} \cos^{2} θ - x y . \sin 2 θ) = a^{2} + b^{2}$

$\Rightarrow (\cos^{2} θ + \sin^{2} θ) x^{2} + (\sin^{2} θ + \cos^{2} θ) y^{2} = a^{2} + b^{2}$

$\Rightarrow (1) x^{2} + (1) y^{2} = a^{2} + b^{2}$

$\Rightarrow x^{2} + y^{2} = a^{2} + b^{2}$

Hence, $x \cos θ + y \sin θ = a$ and $x \sin θ - y \cos θ = b$ are the parametric equations of a circle for all $θ$ satisfying $0 \leq θ < 2 π$ .

Prove that $x \cos θ + y \sin θ = a$ and $x \sin θ - y \cos θ = b$ are the parametric equations of a circle for all $θ$ satisfying $0 \leq θ < 2 π$

Important Questions on Circle

Learn and Prepare for any exam you want !

Prove that xcosθ+ysinθ=a and xsinθ-ycosθ=b are the parametric equations of a circle for all θ satisfying 0≤θ<2π

Important Questions on Circle

Learn and Prepare for any exam you want !

Prove that $x \cos θ + y \sin θ = a$ and $x \sin θ - y \cos θ = b$ are the parametric equations of a circle for all $θ$ satisfying $0 \leq θ < 2 π$