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 \( 5 \).