Kerala SSLC · Mathematics · Class 10
Tangent Lines: Important Questions with Answers
These are 10 important multiple-choice questions from the Kerala SSLC Mathematics chapter “Tangent Lines” (Kerala SCERT syllabus). Each one shows the correct answer and a short explanation of why it is right. Try answering before you read the answer. ExamSummary has 10 practice questions on this chapter in total.
Q1.If a line is tangent to a circle at point P and O is the center of the circle, what is the measure of angle OPT?
- A)45 degrees
- B)90 degrees
- C)180 degrees
- D)It depends on the radius
Answer: B) 90 degrees
According to the tangent property theorem, the tangent at any point of a circle is perpendicular to the radius through the point of contact. Therefore, the angle between the radius OP and the tangent line PT is always 90 degrees.
Q2.From a point located strictly inside a circle, how many tangents can be drawn to that circle?
- A)One
- B)Two
- C)Three
- D)None
Answer: D) None
A tangent line must touch the circle at exactly one point without entering the interior. Since a point inside the circle lies within the boundary, any line passing through it will either intersect the circle at two points or not touch it tangentially.
Q3.The length of a tangent drawn from an external point P to a circle is 12 cm. If the distance from P to the center O is 13 cm, what is the radius of the circle?
- A)5 cm
- B)7 cm
- C)10 cm
- D)25 cm
Answer: A) 5 cm
Using the Pythagorean theorem in triangle OPT (where T is the point of contact), we have OP² = OT² + PT². Substituting the values gives 13² = r² + 12², which simplifies to 169 = r² + 144, resulting in r = 5 cm.
Q4.Two tangents PA and PB are drawn from an external point P to a circle with center O. If angle APB is 80 degrees, what is the measure of angle AOB?
- A)80 degrees
- B)100 degrees
- C)120 degrees
- D)140 degrees
Answer: B) 100 degrees
In the quadrilateral OAPB, angles at A and B are each 90 degrees because radii are perpendicular to tangents. The sum of angles in a quadrilateral is 360 degrees, so angle AOB = 360 - (90 + 90 + 80), which equals 100 degrees.
Q5.What is the maximum number of parallel tangents a single circle can have?
- A)One
- B)Two
- C)Four
- D)Infinite
Answer: B) Two
A circle can only have two parallel tangents, which occur at the endpoints of a diameter perpendicular to the direction of the parallel lines. Any other line parallel to these would either intersect the circle or miss it entirely.
Q6.Which of the following statements regarding tangents drawn from an external point to a circle is true?
- A)Their lengths are unequal
- B)They make different angles with the line joining the center to the external point
- C)Their lengths are equal
- D)They intersect at the center
Answer: C) Their lengths are equal
A fundamental theorem states that the lengths of two tangents drawn from an external point to a circle are equal. This symmetry ensures that triangles formed by the center, external point, and points of contact are congruent.
Q7.Consider two concentric circles with different radii. Can a line be tangent to the smaller circle while lying completely outside the larger circle?
- A)Yes, always
- B)No, never
- C)Yes, only if the circles share a center
- D)Yes, but only at one point
Answer: B) No, never
Since the circles are concentric, the smaller circle is entirely inside the larger circle. A tangent to the smaller circle must pass through its circumference, meaning it will inevitably enter the region occupied by the larger circle, making it impossible to stay completely outside.
Q8.A tangent intersects a circle at _____ point(s).
- A)zero
- B)one
- C)two
- D)three
Answer: B) one
By definition, a tangent line to a curve intersects the curve at exactly one point, known as the point of contact. If it intersected at more than one point, it would be classified as a secant line.
Q9.In the quadrilateral formed by the center of a circle, two points of contact, and an external point where tangents meet, what is the sum of the angles at the center and the external point?
- A)90 degrees
- B)180 degrees
- C)270 degrees
- D)360 degrees
Answer: B) 180 degrees
Since the angles at the points of contact are both 90 degrees, their sum is 180 degrees. Consequently, for the total sum of angles in the quadrilateral to be 360 degrees, the remaining two angles (at the center and the external point) must also sum to 180 degrees.
Q10.For a line to be a tangent to a circle, the perpendicular distance from the center of the circle to the line must be:
- A)Less than the radius
- B)Greater than the radius
- C)Equal to the radius
- D)Twice the radius
Answer: C) Equal to the radius
The defining condition for tangency is that the line touches the circle at exactly one point. Geometrically, this occurs when the shortest distance from the center to the line (the perpendicular distance) is exactly equal to the radius length.