Question

show that the circles x² + y²-8x-6y+21 is orthogonal to the circle x²+y²-2y-15=0​

Answer 1

Step-by-step explanation:

For two circles to be orthogonal ,

g1g2+f1f2=2(c1+c2)

-8×0+-6×-2=12.

2(c1+c2)=2×(6)=12.

So the circles are orthogonal.

Proved

Answer 2

➼Question:-

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Show that the circles x² + y²  - 8x + 6y + 21 = 0 and x² + y²  - 2y - 15 = 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 + 21 = 0

➨2g₁ = -8       2f₁ = -6         c₁ = 21

➨ g₁ = -4         f₁ = -3

➨x² + y²  - 2y - 15 = 0

➨2g₂ = 0       2f₂ = -2         c₂ = -15

➨ g₂ = 0       f₂ = -1

Condition for orthogonal is

2 g₁ g₂ + 2 f₁ f₂ = c₁ + c₂

➨2 (-4) (0) + 2 (-3) (-1) = 21 - 15

➨0 + 6  = 6

     6  = 6

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