A circle having centre $c_1$, radius $r_1$ = 5 cm is placed against a right angle. Another smaller circle having centre $c_2$, radius $r_2$ is also placed touching the sides of angle and the bigger circle as shown in the figure. Find the radius, $r_2$ in cm, of the smaller circle.
To solve this problem, we will use the concept of tangent circles and geometry related to right-angled triangles.
The larger circle with center \( c_1 \) has a radius \( r_1 = 5 \) cm. The smaller circle with center \( c_2 \) is tangent to both sides of the right angle and the larger circle.
We need to find the radius \( r_2 \) of the smaller circle. Based on the given setup and the figure, the centers of the two circles, \( c_1 \) and \( c_2 \), and the point where the smaller circle touches one of the axes form a right triangle.
Let the side length of the right triangle be \( r_1 + r_2 \) because \( c_1c_2 = r_1 + r_2 \) (as \( c_1 \) to \( c_2 \) is the tangent line from \( c_2 \) to the larger circle).
Then, using the Pythagorean theorem in the right-angle triangle formed, we have:
- The larger circle's center lies \( r_1 \) distance away from both axes, forming the "legs" of the triangle.
- The formula for the distance between the centers (i.e., the hypotenuse) would be \( r_1 + r_2 \).
Thus, applying the Pythagorean theorem:
\(c_1c_2 = \sqrt{(r_1 - r_2)^2 + (r_1 - r_2)^2}\)
Now, solving for \( r_2 \):
\(r_1 + r_2 = \sqrt{2}(r_1 - r_2)\)
Rearranging the terms gives:
\(r_1 + r_2 = \sqrt{2}r_1 - \sqrt{2}r_2\)
Solving for \( r_2 \):
- Collecting like terms: \(r_2 + \sqrt{2}r_2 = \sqrt{2}r_1 - r_1\)
- \(r_2(1 + \sqrt{2}) = r_1(\sqrt{2} - 1)\)
- \(r_2 = \frac{r_1(\sqrt{2} - 1)}{1 + \sqrt{2}}\)
- Substitute \( r_1 = 5 \) cm: \(r_2 = \frac{5(\sqrt{2} - 1)}{1 + \sqrt{2}}\)
- Rationalize the denominator: \(r_2 = 5(\sqrt{2} - 1) \times \frac{1 - \sqrt{2}}{1 - \sqrt{2}} = 5(2 - \sqrt{2} - 1 + 1) = 5(3 - 2\sqrt{2})\)
The radius of the smaller circle is \( r_2 = 5(3 - 2\sqrt{2}) \) cm.
Similar Questions
- PQ is a diameter of a circle whose centre is O. If a point R lies on the circle and $\angle RPO$ is $39^\circ$, then find the measure of $\angle RQP$.
- The points A, B and C lie on a circle with centre O. The $\angle \text{ACB} = 47.5^{\circ}$. What is the $\angle \text{AOB}$ on the minor $\widehat{\text{AB}}$?
- $1^{\circ}$ of latitude is equal to ___ km approximately.
- A wheel makes 1000 revolutions in covering a distance of 154 km. Find the radius (in metre) of the wheel.
- Two intersecting circles are said to be orthogonal if the angle between them is a/an _____.
In the given circle with centre O, the obtuse angle at the centre measures $104^\circ$. In the quadrilateral drawn inside the circle, what is the measure of the angle opposite to $\angle\text{O}$? [Figure is not drawn to scale.]
- The chord of contact of tangents drawn from a point on the circle $x^2 + y^2 = a^2$ to the circle $x^2 + y^2 = b^2$ touches the circle $x^2 + y^2 = c^2$ such that $b^m = a^n c^p$, where $m,n,p \in \mathbb{N}$. Find the value of $m + n + p + 10$.
- Find the length of the tangent from any point on the circle $x^2 + y^2 + 2011x + 2012y + 2013 = 0$ to the circle $x^2 + y^2 + 2011x + 2012y + 2014 = 0$.
- If a point (1, 2) is translated 2 units through the positive direction of x-axis and then the tangents drawn from that point to the circle $x^2 + y^2 = 9$, find the angle between the tangents.
- The angle of a sector is 30°. If its radius is 42 cm then the length of the arc of the sector is:
Important Questions from Circles
If 3x + y - 5 = 0 is the equation of a chord of the circle x2 + y2 - 25 = 0, then what are the coordinates of the mid-point of the chord ?
What is the area of minor segment ?
What is the area of major segment ?
A straight line x = y + 2 touches the circle 4(x 2+ y 2) = r 2. The value of r is
If the centre of the circle passing through the origin is (3, 4), then the intercepts cut off by the circle on x-axis and y-axis respectively are
