Question

convert the equation x²-y²=a² in polar equation​

Answer 1

Answer:In polar cordinates

x = r cos(theta)

y = r sin(theta)

the equation becomes

r^2 cos^2theta + r^2 sin^2theta = a^2

2 r^2 = a^2

r = a^2/ 2

Answer 2

x² - y² = a² in polar equation is r² cos2θ = a²

Given :

The equation x² - y² = a²

To find :

To convert in polar equation

Solution :

Step 1 of 3 :

Write down the given equation

Here the given cartesian equation is

x² - y² = a²

Step 2 of 3 :

Write down the transformation

We use the transformation x = r cos θ , y = r sin θ so that a point (x, y) in cartesian coordinate system transformed into the point (r, θ) in polar coordinate system

Step 3 of 3 :

Convert in polar equation

Using the transformation we get

$\displaystyle \sf{ {x}^{2} - {y}^{2} = {a}^{2} }$

$\displaystyle \sf{ \implies {(r \: cos \theta)}^{2} - {(r \: sin \theta)}^{2} = {a}^{2} }$

$\displaystyle \sf{ \implies {r}^{2} { \: cos }^{2} \theta - {r}^{2} { \: sin }^{2} \theta = {a}^{2} }$

$\displaystyle \sf{ \implies {r}^{2}( { \: cos }^{2} \theta - { \: sin }^{2} \theta) = {a}^{2} }$

$\displaystyle \sf{ \implies {r}^{2}cos 2\theta = {a}^{2} }$

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