CAT 1999 — QA Question 52
Directions: Each question is followed by two statements I and II. Mark:
1. if the question can be answered by any one of the statements alone, but cannot be answered by using the other statement alone.
2. if the question can be answered by using either statement alone.
3. if the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
4. if the question cannot be answered even by using both the statements together.
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.
I. The coordinates of D are given.
II. The X-axis bisects one of the tangents.
Answer & solution
Correct answer: 2
- A
1
2
- C
3
- D
4
r and d are given
From statement I, when co-ordinates of D are given,
only one pair of tangents can be drawn onto the given circle from D. So angle made by x-axis for each can be found out.
Hence, statement I alone is sufficient.
Consider statement II. Let the x-axis bisect the tangent QD, i.e. QA = AD.
Here QA = QD = . So using trigonometric ratios in right ΔCQA, we can determine ∠CAQ. Therefore, ∠DAB is equal to ∠CAQ CAQ (vertically opposite angles).
Consider the other tangent DP. Let it intersect x-axis at point L.
∠CDQ can be determined using trigonometric ratios (as two of the sides are given in right ΔCDQ). Also, ∠CDQ is equal to ∠CDP (since the two right ΔCQD and ΔCPD are congruent). Drop a perpendicular DB on x-axis. In right ΔDBL, we can find ∠BDL = 180o − (∠ADB + ∠CDQ + ∠CDP)
= 180° − 2∠CDQ − ∠ADB . Applying angle sum property of triangle, we can determine ∠DLB.
Hence, statement II alone is sufficient.