Question

If \( P \left( \frac{\pi}{3} \right) \) and \( Q \left( \frac{2\pi}{3} \right) \) represent two points on the circle \( x^2 + y^2 - 4x - 6y - 12 = 0 \) in parametric form, then the length of the chord PQ is:

• A. \( 4\sqrt{3} \)
• B. \( 5 \)
• C. \( 5\sqrt{2} \)
• D. \( 13 \)

Hint:

When working with parametric equations of a circle, use the center and radius to find the coordinates of points and apply the distance formula to find the length of the chord.

Answer 1

The Correct Option is B

We are given the circle equation \( x^2 + y^2 - 4x - 6y - 12 = 0 \). First, we rewrite the equation in standard form by completing the square: \[ (x - 2)^2 + (y - 3)^2 = 25 \] This represents a circle with center \( (2, 3) \) and radius 5. Step 1: Parametric equations for points on the circle are: \[ P \left( \frac{\pi}{3} \right) = (2 + 5\cos(\frac{\pi}{3}), 3 + 5\sin(\frac{\pi}{3})) \] \[ Q \left( \frac{2\pi}{3} \right) = (2 + 5\cos(\frac{2\pi}{3}), 3 + 5\sin(\frac{2\pi}{3})) \] Step 2: Find the distance between points \( P \) and \( Q \): \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substitute the parametric coordinates of \( P \) and \( Q \) into this formula. After calculation, the length of the chord \( PQ \) is \( 5 \). % Final Answer The length of the chord \( PQ \) is \( 5 \).

Answer 2

Step 1: Convert the circle equation to standard form
The given circle equation is:
\[ x^2 + y^2 - 4x - 6y - 12 = 0 \]
Group the terms:
\[ (x^2 - 4x) + (y^2 - 6y) = 12 \]
Complete the square:
\[ (x^2 - 4x + 4) + (y^2 - 6y + 9) = 12 + 4 + 9 \Rightarrow (x - 2)^2 + (y - 3)^2 = 25 \]
So, the circle has center at \( (2, 3) \) and radius \( R = \sqrt{25} = 5 \)

Step 2: Parametric form of a point on a circle
For a circle with center \( (h, k) \) and radius \( R \), the parametric coordinates of a point are:
\[ x = h + R \cos \theta,\quad y = k + R \sin \theta \]
Substitute \( h = 2 \), \( k = 3 \), \( R = 5 \)

For point \( P\left( \frac{\pi}{3} \right) \):
\[ x_P = 2 + 5 \cos\left( \frac{\pi}{3} \right) = 2 + 5 \cdot \frac{1}{2} = 4.5 \] \[ y_P = 3 + 5 \sin\left( \frac{\pi}{3} \right) = 3 + 5 \cdot \frac{\sqrt{3}}{2} = 3 + \frac{5\sqrt{3}}{2} \]
For point \( Q\left( \frac{2\pi}{3} \right) \):
\[ x_Q = 2 + 5 \cos\left( \frac{2\pi}{3} \right) = 2 + 5 \cdot \left( -\frac{1}{2} \right) = -0.5 \] \[ y_Q = 3 + 5 \sin\left( \frac{2\pi}{3} \right) = 3 + 5 \cdot \frac{\sqrt{3}}{2} = 3 + \frac{5\sqrt{3}}{2} \]

Step 3: Use distance formula to find chord length PQ
Note that both points have the same \( y \)-coordinate, so PQ is a horizontal chord:
\[ PQ = |x_P - x_Q| = |4.5 - (-0.5)| = 5 \]

Final Answer:
5