Question

If power of a point \( (4,2) \) with respect to the circle \( x^2 + y^2 - 2x + 6y + a^2 - 16 = 0 \) is 9, then the sum of the lengths of all possible intercepts made by such circles on the coordinate axes is 

• A. \( 16 + 4\sqrt{6} \)
• B. \( 16 + 4\sqrt{6} - 6\sqrt{2} \)
• C. \( 16 + 4\sqrt{6} + 6\sqrt{2} \)
• D. \( 16 + 6\sqrt{2} \)

Hint:

To compute the sum of intercepts made by a circle on coordinate axes, use the general formula \( 2(\sqrt{g^2 + f^2 - c}) \) and substitute the known values.

Answer 1

The Correct Option is A

Step 1: Standard Equation of a Circle
The general equation of a circle is given as: \[ x^2 + y^2 + 2gx + 2fy + c = 0. \] Comparing with the given equation: \[ x^2 + y^2 - 2x + 6y + a^2 - 16 = 0, \] we identify: \[ g = -1, \quad f = 3, \quad c = a^2 - 16. \] Step 2: Compute the Power of the Point
The power of a point \( (x_1, y_1) \) with respect to a circle is given by: \[ P = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c. \] Substituting \( (x_1, y_1) = (4,2) \): \[ P = 4^2 + 2^2 + 2(-1)(4) + 2(3)(2) + a^2 - 16. \] \[ P = 16 + 4 - 8 + 12 + a^2 - 16. \] \[ P = a^2 + 8. \] Given that \( P = 9 \), we solve for \( a^2 \): \[ a^2 + 8 = 9 \Rightarrow a^2 = 1. \] Step 3: Find the Sum of Intercepts
The sum of the intercepts made by the circle on the coordinate axes is given by: \[ 2\left( \sqrt{g^2 + f^2 - c} + \sqrt{g^2 + f^2 - c} \right). \] Substituting values: \[ \sqrt{(-1)^2 + (3)^2 - (1 - 16)} = \sqrt{1 + 9 + 15} = \sqrt{25} = 5. \] Thus, the sum of the intercepts is: \[ 2(8 + 2\sqrt{6}) = 16 + 4\sqrt{6}. \] Final Answer: \[ \boxed{16 + 4\sqrt{6}}. \]

Answer 2

To solve the problem, we start by examining the equation of the circle given by \(x^2 + y^2 - 2x + 6y + a^2 - 16 = 0\). We rewrite it in the standard form by completing the square:

\(x^2 - 2x + y^2 + 6y = 16 - a^2\).

Completing the square for \(x\) and \(y\), we have:

\((x-1)^2 - 1 + (y+3)^2 - 9 = 16 - a^2\).

Rearranging gives:

\((x-1)^2 + (y+3)^2 = a^2 + 6\).

Thus, the center of the circle is \((1, -3)\) and radius \(r = \sqrt{a^2 + 6}\).

The power of a point \((4, 2)\) with respect to the circle is:

\((4-1)^2 + (2+3)^2 - r^2 = 9\).

Calculating the left side:

\(3^2 + 5^2 - (a^2 + 6) = 9\).

Simplify:

\(9 + 25 - a^2 - 6 = 9\).

This simplifies to:

\(28 - a^2 = 9\).

Thus:

\(a^2 = 19\).

The circle's equation becomes:

\((x-1)^2 + (y+3)^2 = 25\), which implies \(r = 5\).

The intercepts on the x-axis occur at:

\(y = 0\). Substituting in:\((x-1)^2 + 9 = 25\), we find:\( (x-1)^2 = 16\), thus \(x = 5\) or \(x = -3\).

Length of intercept on x-axis is \(|5 + 3| = 8\).

The intercepts on the y-axis occur at:

\(x = 0\). Substituting in:\(1 + (y+3)^2 = 25\), we find:\((y+3)^2 = 24\), thus \(y = \sqrt{24} - 3\) or \(y = -\sqrt{24} - 3\).

The length of intercept on the y-axis is \(|\sqrt{24} - 3 + \sqrt{24} + 3| = 2\sqrt{24} = 4\sqrt{6}\).

The total sum of the lengths of intercepts is:

\(8 + 4\sqrt{6}\), which is \(16 + 4\sqrt{6}\).

Thus, the sum of the lengths of all possible intercepts made by such circles on the coordinate axes is:
\(16 + 4\sqrt{6}\).