Question

Find k if the circles + y2 – 5x – 14y - 34 = 0 and x2 + y2 + 2x + 4y + k = 0
are orthogonal.
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Answer 1

We have to find the value of k for which circles x² + y² - 5x - 14y - 34 = 0 and x² + y² + 2x + 4y + k = 0 are orthogonal.

To find : condition of orthogonality of circles is given by, 2g₁g₂ + 2f₁f₂ = c₁ + c₂

Here equations of circle are ;

x² + y² - 5x - 14y - 34 = 0

x² + y² + 2x + 4y + k = 0

so, g₁ = -5/2, f₁ = -7 and c₁ = -34

g₂ = 1, f₂ = 2 and c₂ = k

Now 2g₁g₂ + 2f₁f₂ = c₁ + c₂

⇒2(-5/2)(1) + 2(-7)(2) = -34 + k

⇒-5 - 28 = -34 + k

⇒-33 + 34 = k

⇒k = 1

Therefore the value of k is 1

Answer 2

Answer:-

We have to find the value of k for which circles x² + y² - 5x - 14y - 34 = 0 and x² + y² + 2x + 4y + k = 0 are orthogonal.

To find :

condition of orthogonality of circles is given by, 2g₁g₂ + 2f₁f₂ = c₁ + c₂

Here equations of circle are ;

x² + y² - 5x - 14y - 34 = 0

x² + y² + 2x + 4y + k = 0

so, g₁ = -5/2, f₁ = -7 and c₁ = -34

g₂ = 1, f₂ = 2 and c₂ = k

Now

2g₁g₂ + 2f₁f₂ = c₁ + c₂

⇒2(-5/2)(1) + 2(-7)(2) = -34 + k

⇒-5 - 28 = -34 + k

⇒-33 + 34 = k

⇒k = 1

Therefore the value of k is 1

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