Question:
A circle passes through the points \( (0, 0) \) and \( (0, 1) \) and also touches the circle \( x^2 + y^2 = 16 \). The radius of the circle is:
A circle passes through the points \( (0, 0) \) and \( (0, 1) \) and also touches the circle \( x^2 + y^2 = 16 \). The radius of the circle is:
Show Hint
When a circle is defined by points on an axis, use the symmetry of the perpendicular bisector to reduce the number of variables immediately.
Updated On: May 6, 2026
- \( 1 \)
- \( 2 \)
- \( 3 \)
- \( 4 \)
- \( 5 \)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
A circle passing through \( (0,0) \) and \( (0,1) \) must have its center on the perpendicular bisector of the segment joining these points, which is the line \( y = 1/2 \). Let the center be \( (h, 1/2) \). The radius \( r \) will be the distance from the center to \( (0,0) \).
Step 1: Define the radius squared.
\[ r^2 = h^2 + (1/2)^2 = h^2 + 1/4 \]
Step 2: Use the condition of touching circles.
The circle touches \( x^2 + y^2 = 4^2 \) (radius 4, center (0,0)).
If they touch internally, the distance between centers is
\( |4 - r| \). Distance between \( (0,0) \) and \( (h, 1/2) \) is \( \sqrt{h^2 + 1/4} \),
which is just \( r \).
So, \( r = |4 - r| \).
Case 1: \( r = 4 - r \Rightarrow 2r = 4 \Rightarrow r = 2 \).
Step 1: Define the radius squared.
\[ r^2 = h^2 + (1/2)^2 = h^2 + 1/4 \]
Step 2: Use the condition of touching circles.
The circle touches \( x^2 + y^2 = 4^2 \) (radius 4, center (0,0)).
If they touch internally, the distance between centers is
\( |4 - r| \). Distance between \( (0,0) \) and \( (h, 1/2) \) is \( \sqrt{h^2 + 1/4} \),
which is just \( r \).
So, \( r = |4 - r| \).
Case 1: \( r = 4 - r \Rightarrow 2r = 4 \Rightarrow r = 2 \).
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Circle Questions
- The area of the circle $x^2 - 2x + y^2 - 10\,y + k = 0$ is $25 \pi $ . The value of k is equal to
- The equation of the circle that can be inscribed in the square formed by \(x^2-8x+12=0\) and \(y^2-14y+45=0\) is ?
- For the circle \(C=x^2+y^2-6x+2y=0\),which of the following is incorrect:
- Suppose the line joining distinct points P and Q on \((x-2)^{2}+(y-1)^{2}=r^{2}\) is the diameter of \( (x-1)^2+(y-3)^2=4\).The value of r is
For \(i=1,2,3,4\), suppose the points\((\cosθi,\secθi)\) lie on the boundary of a circle,where \(θi∈[0,\frac{π}{6})\) are distinct. Then \(\cosθ_1\ \cosθ_2\ \cosθ_3\ \cosθ_4\) equals
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions