Question

Show that the circles x² + y²  - 8x + 6y - 23 = 0 and x² + y²  - 2x - 5y + 16 = 0 are orthogonal.​

Answer 1

➼Question:-

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•Show that the circles x² + y²  - 8x + 6y - 23 = 0 and x² + y²  - 2x - 5y + 16 = 0 are orthogonal.

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➼Solution:-

First let us find the values of g₁, g₂,f₁ and f₂ from the equations by comparing the given equation with the general form of the circle

➨x² + y² + 2gx + 2fy + c = 0

⇒ x² + y²  - 8x + 6y - 23 = 0

⇒ 2g₁ = -8       2f₁ = 6         c₁ = -23

⇒  g₁ = -4         f₁ = 3

⇒ x² + y²  - 2x - 5y + 16 = 0

⇒ 2g₂ = -2       2f₂ = -5         c₂ = 16

⇒  g₂ = -1       f₂ = -5/2

Condition for orthogonal is

➨2 g₁ g₂ + 2 f₁ f₂ = c₁ + c₂

⇒ 2 (-4) (-1) + 2 (3) (-5/2) = -23 + 16

⇒ 8 - 15  = -7

⇒      - 7 = -7

So the given condition is satisfied.So the given circles are orthogonal.

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