The number of common tangerts to
the circles x²+y²_4X-6y - 12=0
and x² + y² + 6x + 18y+26=0 is
(ค)
3
B 4
1
D
2
Answers
EXPLANATION.
Number of common tangent to the circle.
⇒ x² + y² - 4x - 6y - 12 = 0. - - - - - (1).
⇒ x² + y² + 6x + 18y + 26 = 0. - - - - - (2).
As we know that,
General equation of the circle.
⇒ x² + y² + 2gx + 2fy + c = 0.
From equation (1), we get.
⇒ x² + y² - 4x - 6y - 12 = 0. - - - - - (1).
⇒ Centre of circle = (-g,-f).
⇒ Centre of circle = (2,3).
⇒ Radius of circle = √g² + f² - c.
⇒ Radius of circle = √(2)² + (3)² - (-12).
⇒ Radius of circle = √4 + 9 + 12.
⇒ Radius of circle = √25.
⇒ Radius of circle = 5.
From equation (2), we get.
⇒ x² + y² + 6x + 18y + 26 = 0. - - - - - (2).
⇒ Centre of circle = (-g,-f).
⇒ Centre of circle = (-3,-9).
⇒ Radius of circle = √g² + f² - c.
⇒ Radius of circle = √(-3)² + (-9)² - 26.
⇒ Radius of circle = √9 + 81 - 26.
⇒ Radius of circle = √90 - 26.
⇒ Radius of circle = √64.
⇒ Radius of circle = 8.
⇒ C₁C₂ = √(x₁ - x₂)² + (y₁ - y₂)².
⇒ C₁ = (2,3) & C₂ = (-3,-9).
⇒ C₁C₂ = √[2 - (-3)]² + [3 - (-9)]².
⇒ C₁C₂ = √(2 + 3)² + (3 + 9)².
⇒ C₁C₂ = √(5)² + (12)².
⇒ C₁C₂ = √25 + 144.
⇒ C₁C₂ = √169.
⇒ C₁C₂ = 13.
⇒ r₁ + r₂ = 5 + 8.
⇒ r₁ + r₂ = 13.
As we know that,
⇒ C₁C₂ = r₁ + r₂.
It means 3 common tangents.
Option [A] is correct answer.
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