Math, asked by vishnuvardhangudepu, 1 month ago

*7. Show that the circles ra+y2- 4r - 6y - 12 = 0
and r2 + y2 + 6x + 18y + 26 = 0 touch each
other. Find the point of contact and the
equation of common tangent at the point of
contact. (Mar-2017(AP), 2013, 2008, 2005, 2002)​

Answers

Answered by amansharma264
4

EXPLANATION.

Equation of circle,

⇒ C₁ = x² + y² - 4y - 6y - 12 = 0.

⇒ C₂ = x² + y² + 6x + 18y + 26 = 0.

As we know that,

General equation of circle,

⇒ x² + y² + 2gx + 2fy + c = 0.

Compare both the equation with general equation, we get.

Centre of circle = (-g,-f).

⇒ Radius of the circle = √g² + f² - c.

⇒ Centre of circle = C₁ = (2,3).

⇒ Radius of circle = r₁ = √(2)² + (3)² - (-12).

⇒ Radius of circle = r₁ = √4 + 9 + 12.

⇒ Radius of circle = r₁ = √25.

⇒ Radius of circle = r₁ = 5.

⇒ Centre of circle = C₂ = (-3,-9).

⇒ Radius of circle = r₂ = √(-3)² + (-9)² - 26.

⇒ Radius of circle = r₂ = √9 + 81 - 26.

⇒ Radius of circle = r₂ = √90 - 26.

⇒ Radius of circle = r₂ = √64.

⇒ Radius of circle = r₂ = 8.

As we know that,

⇒ r₁ + r₂ = 5 + 8 = 13.

⇒ 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.

As we know that,

⇒ C₁C₂ = r₁ + r₂.

(1) = No of direct common tangents = 2.

(2) = No of transverse common tangents = 1.

(3) = No of total common tangents = 3.

As we know that,

Formula of :

Section formula for internal division.

⇒ mx₂ + nx₁/m + n & my₂ + ny₁/m + n.

Section formula for external division.

⇒ mx₂ - nx₁/m - n & my₂ - ny₁/m - n.

⇒ m = r₁ = 5.

⇒ n = r₂ = 8.

⇒ C₁ = (2,3).

⇒ C₂ = (-3,-9).

⇒ x₁ = (5)(-3) + (8)(2)/5 + 8.

⇒ - 15 + 16/13.

⇒ 1/13

⇒ y₁ = (5)(-9) + (8)(3)/5 + 8.

⇒ y = - 45 + 24/13.

⇒ y = - 21/13.

Their Co-ordinates = (1/13, -21/13).

⇒ x₂ = (5)(-3) - (8)(2)/5 - 3.

⇒ x = - 15 - 16/-3.

⇒ x = -31/-3.

⇒ x = 31/3.

⇒ y₂ = (5)(-9) - (8)(3)/5 - 8.

⇒ y = - 45 - 24/-3.

⇒ y = - 69/-3.

⇒ y = 23.

Their Co-ordinates = (31/3, 23).

As we know that,

Equation of common tangents = S₁ - S₂ = 0.

⇒ x² + y² - 4x - 6y - 12 - (x² + y² + 6x + 18y + 26) = 0.

⇒ x² + y² - 4x - 6y - 12 - x² - y² - 6x - 18y - 26 = 0.

⇒ - 4x - 6y - 12 - 6x - 18y - 26 = 0.

⇒ - 10x - 24y - 38 = 0.

⇒ -2(5x + 12y + 19) = 0.

⇒ 5x + 12y + `19 = 0.

                                                                                                                                     

MORE INFORMATION.

Relative position of two circles and no of common tangents.

Let C₁ (h₁, k₁) and C₂ (h₂, k₂) be the Centre  of two circle and r₁, r₂ be their radius then,

(1) = C₁C₂ > r₁ + r₂ ⇒ do not intersect or one outside the other ⇒ 4 common tangents.

(2) = C₁C₂ < | r₁ - r₂ | ⇒ one inside the other ⇒ 0 common tangents.

(3) = C₁C₂ = r₁ + r₂ ⇒ external touch ⇒ 3 common tangents.

(4) = C₁C₂ = | r₁ - r₂ | ⇒ internal touch ⇒ 1 common tangents.

(5) = | r₁ - r₂ | < C₁C₂ < r₁ + r₂ ⇒ intersection at two real points. ⇒ 2 common tangents.

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