*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
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.