Question

There is a circle with centre C at the origin and radius r cm. Two tangents are drawn from an external point D at a distance d cm from the centre. What are the angles between each tangent and the X-axis?
A. The coordinates of D are given.
B. The X-axis bisects one of the tangents.

• A. The question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
• B. The question can be answered by using both statements together, but cannot be answered using either statement alone.
• C. The question can be answered by using either statement alone.
• D. The question cannot be answered even by using both statements together.

Hint:

Decide for each statement whether it fixes the position of D relative to the X-axis (up to reflection), since r and d are already known.

Answer 1

The Correct Option is C

Step 1: Statement A alone: With D known and r known, the two tangent lines from D are fixed (their slopes are the roots of \((mp-q)^2=r^2(1+m^2)\) for \(D=(p,q)\)), so both angles with the X-axis can be calculated. Sufficient.

Step 2: Statement B alone: Let T be the point of contact and M the midpoint of tangent segment TD, lying on the X-axis. Tangent length \(t=\sqrt{d^2-r^2}\). In the right triangle CTM (right angle at T), \(CT=r\) and \(TM=t/2\), so the angle between the tangent and the X-axis (line CM) satisfies \(\tan\alpha=\dfrac{2r}{\sqrt{d^2-r^2}}\). The position of D is therefore fixed up to reflection in the X-axis, so the angle of the other tangent is also determined (e.g. for r = 1, d = 2: about 49.1 degrees and 10.9 degrees). Sufficient.

Final Answer:
The correct option is (3): The question can be answered by using either statement alone.