Question

The equation of the circle touching the x-axis at (5, 0) and the line y = 10 is

• A. x²+y²-10x-10y+25=0
• B. x²+y²-10x-10y-25=0
• C. x²+y²-5x-5y-5=0
• D. x²+y²-5x-5y+5=0
• E. x²+y²+10x+10y-25=0

Answer 1

The Correct Option is A

The general equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the center of the circle and \( r \) is the radius. 1. The center of the circle is on the line \( y = 10 \) because the circle touches this line. So, the center has coordinates \( (h, 10) \). 2. The circle touches the x-axis at \( (5, 0) \), meaning the distance from the center \( (h, 10) \) to the x-axis is equal to the radius of the circle. This gives the equation: \[ r = 10 \] 3. The distance from the center of the circle \( (h, 10) \) to the point \( (5, 0) \) is also equal to the radius. Using the distance formula: \[ r = \sqrt{(h - 5)^2 + (10 - 0)^2} \] Since \( r = 10 \), we get: \[ 10 = \sqrt{(h - 5)^2 + 100} \] Squaring both sides: \[ 100 = (h - 5)^2 + 100 \] Simplifying: \[ 0 = (h - 5)^2 \] Thus, \( h = 5 \). Now, we can write the equation of the circle with center \( (5, 10) \) and radius 10: \[ (x - 5)^2 + (y - 10)^2 = 10^2 \] Expanding: \[ x^2 - 10x + 25 + y^2 - 20y + 100 = 100 \] Simplifying: \[ x^2 + y^2 - 10x - 10y + 25 = 0 \]

The correct option is (A) : \(x^2+y^2-10x-10y+25=0\)

Answer 2

Since the circle touches the x-axis at (5, 0), the center of the circle must be (5, r), where r is the radius of the circle.

Since the circle also touches the line y = 10, the distance from the center (5, r) to the line y = 10 must be equal to the radius r. Therefore, \(|r - 10| = r\).

We have two cases:

• Case 1: \(r - 10 = r\). This simplifies to -10 = 0, which is impossible.
• Case 2: \(r - 10 = -r\). This simplifies to \(2r = 10\), so \(r = 5\).

So, the center of the circle is (5, 5) and the radius is 5.

The equation of a circle with center (h, k) and radius r is \((x - h)^2 + (y - k)^2 = r^2\).

In this case, the center is (5, 5) and the radius is 5, so the equation is \((x - 5)^2 + (y - 5)^2 = 5^2\).

Expanding the equation, we get:

\(x^2 - 10x + 25 + y^2 - 10y + 25 = 25\)

\(x^2 + y^2 - 10x - 10y + 50 = 25\)

\(x^2 + y^2 - 10x - 10y + 25 = 0\)

Therefore, the equation of the circle is \(x^2 + y^2 - 10x - 10y + 25 = 0\).