Question

If \( p \) and \( q \) be the longest and the shortest distance respectively of the point (-7,2) from any point (\(\alpha, \beta\)) on the curve whose equation is \[ x^2 + y^2 - 10x - 14y - 51 = 0 \] then the geometric mean (G.M.) of \( p \) is:

• A. \( 2\sqrt{11} \)
• B. \( 5\sqrt{5} \)
• C. 13
• D. 11

Hint:

Understanding the concept of distance from a point to a circle helps in solving such problems efficiently.

Answer 1

The Correct Option is A

The given equation of the curve is: \[ x^2 + y^2 - 10x - 14y - 51 = 0 \] We rewrite it in standard circle form by completing the square. \[ (x^2 - 10x) + (y^2 - 14y) = 51 \] Completing squares: \[ (x - 5)^2 - 25 + (y - 7)^2 - 49 = 51 \] \[ (x - 5)^2 + (y - 7)^2 = 5\sqrt{5}^2 \] So, the center \( C(5,7) \) and radius \( r = 5\sqrt{5} \). Now, the distance of point \( (-7,2) \) from the center \( C(5,7) \): \[ PC = \sqrt{(5 + 7)^2 + (7 - 2)^2} \] \[ PC = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \] Thus, the farthest and closest distances: \[ p = 13 + 5\sqrt{5}, \quad q = 13 - 5\sqrt{5} \] The geometric mean: \[ \sqrt{pq} = \sqrt{(13 - 5\sqrt{5}) (13 + 5\sqrt{5})} \] Using the identity \( (a - b)(a + b) = a^2 - b^2 \): \[ \sqrt{169 - 125} = \sqrt{44} = 2\sqrt{11} \]

Answer 2

If \( p \) and \( q \) be the longest and the shortest distance respectively of the point (-7,2) from any point (\(\alpha, \beta\)) on the curve whose equation is
\[ x^2 + y^2 - 10x - 14y - 51 = 0 \]
then the geometric mean (G.M.) of \( p \) is:
We first rewrite the equation of the curve in standard form by completing the square:
\[ x^2 - 10x + y^2 - 14y = 51 \] Completing the square for \(x\) and \(y\):
\[ (x - 5)^2 + (y - 7)^2 = 81 \] This is the equation of a circle with center \( (5, 7) \) and radius 9.
Now, we calculate the distance from the point (-7,2) to the center of the circle:
\[ d = \sqrt{(5 - (-7))^2 + (7 - 2)^2} = \sqrt{(5 + 7)^2 + (7 - 2)^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \] The longest distance, \( p \), is the sum of the radius of the circle and the distance from the point to the center: \( p = 13 + 9 = 22 \).
The shortest distance, \( q \), is the difference between the distance from the point to the center and the radius: \( q = 13 - 9 = 4 \).
The geometric mean of \( p \) and \( q \) is:
\[ \text{G.M.} = \sqrt{p \cdot q} = \sqrt{22 \cdot 4} = \sqrt{88} = 2\sqrt{11} \] Thus, the correct answer is:

\( 2\sqrt{11} \)