Question 6 of 11intermediate🔍 AnalyzeLong Answer4 marks

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Correct Answer

Given: A circle with centre O and a tangent XY touching the circle at point P. A line l is perpendicular to XY at the point of contact P.

To Prove: The perpendicular line l passes through the centre O of the circle.

Construction: Join OP (the radius through the point of contact P).

Proof:

From Theorem 10.1, we know that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Since XY is a tangent at point P and OP is the radius through P, we have:

OP⊥XYOP \perp XY

We are given that line l is perpendicular to the tangent XY at point P, i.e.:

l⊥XY at Pl \perp XY \text{ at } P

Now, at a given point P on a line XY, there can be only one perpendicular line. Since both OP and line l are perpendicular to XY at the same point P, they must coincide with each other.

Therefore, line l coincides with OP. Since OP passes through the centre O (being the radius), the perpendicular line l must also pass through the centre O.

∴ Hence, the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Hence Proved.

Exercise: EXERCISE 10.2 | Q: 5 | (Page 9)
For More Understanding

Explanation

This question asks for the converse of Theorem 10.1. Theorem 10.1 states that the radius through the point of contact is perpendicular to the tangent. The converse states that if a line is perpendicular to the tangent at the point of contact, it must be the radius (and hence pass through the centre).

The proof uses the uniqueness of a perpendicular through a given point on a line — since the radius OP is perpendicular to the tangent at P (by Theorem 10.1), and the given line l is also perpendicular at P, they must be the same line.

This is a standard 4-mark proof question testing understanding of the relationship between tangents and radii.

Solution Steps

  1. Step 1: State Theorem 10.1 - the radius through the point of contact is perpendicular to the tangent

  2. Step 2: Apply the theorem to conclude OP ⊥ XY

  3. Step 3: Note that line l is given perpendicular to XY at P

  4. Step 4: Use the fact that only one perpendicular can be drawn at a point on a line

  5. Step 5: Conclude that l coincides with OP and hence passes through centre O

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