The circles touch internally. The sum of their areas is 116π cm² and distance between their centres is 6 cm. Find the radius of larger circle.
10 cm
Since the circles touch internally, the distance between their centres equals the difference of their radii: \(R-r=6\), where \(R\) is the radius of the larger circle.
The sum of the areas is \(\pi R^2+\pi r^2=116\pi\), so \(R^2+r^2=116\).
Substituting \(R=r+6\): \((r+6)^2+r^2=116\), i.e. \(2r^2+12r+36=116\), so \(2r^2+12r-80=0\), i.e. \(r^2+6r-40=0\).
Solving: \(r=\dfrac{-6\pm\sqrt{36+160}}{2}=\dfrac{-6\pm14}{2}\); taking the positive root, \(r=4\).
So \(R=r+6=10\) cm.
Important Questions from Circles
The sum of the radius and diameter of a circle is 84 cm. What is the circumference of this circle?
The equation of a circle with diameters are 2x - 3y + 12 = 0 and x + 4y - 5 = 0 and area of 154 sq. units is
The internal center of similitude of two circles $ (x - 1)^2 + (y - 3)^2 = 4 $ and $ (x + 5)^2 + (y - 9)^2 = 16 $ is
The inner circumference of a circular race track 14 cm wide is 440 cm. Find the radius of the outer circle.
Three circles touch each other externally. The distance between their centres is 5 cm, 6 cm and 7 cm. Find the radii of the circles-
