Question

The equation of the circle which touches the circle $S \equiv x^2+y^2-10x-4y+19=0$ at the point (2,3) internally and having radius equal to half of the radius of the circle S=0 is

• A. $x^2+y^2+7x+5y+64=0$
• B. $x^2+y^2-7x-5y+16=0$
• C. $x^2+y^2-14x-10y+16=0$
• D. $x^2+y^2-5x-7y+16=0$

Hint:

When two circles with centers $C_1, C_2$ and radii $R_1, R_2$ touch internally, the distance between their centers is the difference of their radii, $C_1C_2 = |R_1-R_2|$. The point of tangency and the two centers are collinear.

Answer 1

The Correct Option is B

Step 1: Find the center and radius of the given circle S.
The equation is $x^2+y^2-10x-4y+19=0$. The center is $C_1 = (-\frac{-10}{2}, -\frac{-4}{2}) = (5, 2)$. The radius is $R_1 = \sqrt{5^2+2^2-19} = \sqrt{25+4-19} = \sqrt{10}$.

Step 2: Determine the center and radius of the required circle S'.
Let the center of S' be $C_2$ and radius be $R_2$. We are given that $R_2 = \frac{1}{2}R_1 = \frac{\sqrt{10}}{2}$. The circles touch internally at the point $P(2,3)$. For internal tangency, the centers and the point of contact are collinear. The center $C_2$ must lie on the line segment joining $C_1$ and $P$. The distance between the centers is $C_1C_2 = R_1 - R_2 = \sqrt{10} - \frac{\sqrt{10}}{2} = \frac{\sqrt{10}}{2}$. Since $C_2P = R_2 = \frac{\sqrt{10}}{2}$, we have $C_1C_2 = C_2P$. This means $C_2$ is the midpoint of the segment $C_1P$.

Step 3: Calculate the coordinates of the center $C_2$.
Using the midpoint formula for $C_1(5,2)$ and $P(2,3)$: \[ C_2 = \left(\frac{5+2}{2}, \frac{2+3}{2}\right) = \left(\frac{7}{2}, \frac{5}{2}\right). \]

Step 4: Write the equation of the required circle S'.
The circle has center $(7/2, 5/2)$ and radius $R_2 = \sqrt{10}/2$. The equation is $(x - h)^2 + (y - k)^2 = R^2$. \[ \left(x - \frac{7}{2}\right)^2 + \left(y - \frac{5}{2}\right)^2 = \left(\frac{\sqrt{10}}{2}\right)^2. \] \[ x^2 - 7x + \frac{49}{4} + y^2 - 5y + \frac{25}{4} = \frac{10}{4}. \] Multiplying by 4 to clear the denominators: \[ 4x^2 - 28x + 49 + 4y^2 - 20y + 25 = 10. \] \[ 4x^2 + 4y^2 - 28x - 20y + 74 - 10 = 0. \] \[ 4x^2 + 4y^2 - 28x - 20y + 64 = 0. \] Dividing by 4 gives the final equation: \[ x^2 + y^2 - 7x - 5y + 16 = 0. \] This matches option (B).