Question

For the circle \( x-2=5\cos\theta \), \( y+1=5\sin\theta \) where \( \theta \) is the parameter, the line \( x=1+\frac{\sqrt{3}}{2}r \), \( y=-2+\frac{r}{2} \) where \( r \) is the parameter, is a

• A. Chord of the circle other than diameter
• B. Tangent of the circle
• C. Diameter of the circle
• D. Line that does not meet the circle

Hint:

Calculating the "Power of the Point" ($S_1$) is the quickest way to determine position. Negative = Inside (Chord), Zero = On circle (Tangent if direction matches, chord otherwise), Positive = Outside (Secant, Tangent, or Non-intersecting).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:

First, identify the circle's center and radius. Then identify a fixed point on the given line and check its position relative to the circle. Finally, check whether the line passes through the center of the circle.

Step 2: Detailed Explanation:

Circle Analysis:

Given:
x - 2 = 5 cos θ
y + 1 = 5 sin θ

Squaring and adding both equations:
(x - 2)2 + (y + 1)2 = 25

So, the circle has:
Center C(2, -1)
Radius R = 5

Line Analysis:

Given:
x = 1 + (√3/2)r
y = -2 + r/2

This is the equation of a straight line passing through the fixed point:
P(1, -2)

Now check the position of point P relative to the circle:

S1 = (1 - 2)2 + (-2 + 1)2 - 25
= 1 + 1 - 25
= -23

Since S1 < 0, the point P lies inside the circle.

Any straight line passing through an interior point of a circle intersects the circle at two distinct points.
Therefore, the line is a chord of the circle.

Check whether it is a diameter:

A chord is a diameter only if it passes through the center of the circle.

Check whether C(2, -1) lies on the line.

Substitute x = 2 into the line equation:
2 = 1 + (√3/2)r
(√3/2)r = 1
r = 2/√3

Now substitute this value of r into the y-equation:
y = -2 + (1/2)(2/√3)
y = -2 + 1/√3
y ≈ -2 + 0.577
y ≈ -1.423

But the center has y-coordinate -1.

Since -1 ≠ -1.423, the center does not lie on the line.

Hence, the line is a chord, but not a diameter.

Step 3: Final Answer:

It is a chord of the circle other than the diameter.