Question

In the given figure, PQ is a tangent to a circle with centre O(–5, 3). If coordinates of P and Q are (3, 1) and (0, 6) respectively, then using distance formula, show that PQ ⊥ OQ.

Answer 1

Given:

Centre O = (–5, 3), P = (3, 1), Q = (0, 6).

PQ is a tangent at Q, so the radius OQ is perpendicular to the tangent at Q.

Step-wise calculation:

1. Use the distance formula d² = (x₂ – x₁)² + (y₂ – y₁)².

2. Compute OQ²: O = (–5, 3), Q = (0, 6) 

OQ² = (0 – (–5))² + (6 – 3)²

= 5² + 3²

= 25 + 9

= 34

3. Compute PQ²: P = (3, 1), Q = (0, 6) 

PQ² = (3 – 0)² + (1 – 6)²

= 3² + (–5)²

= 9 + 25

= 34

4. Compute OP² to apply the converse of Pythagoras: 

O = (–5, 3), P = (3, 1) 

OP² = (3 – (–5))² + (1 – 3)² 

= 8² + (–2)²

= 64 + 4

= 68

5. Observe that OQ² + PQ²

= 34 + 34 

= 68

= OP² 

By the converse of the Pythagorean theorem, triangle OQP is right-angled at Q, so OQ ⟂ PQ.

Since OQ² + PQ² = OP², triangle OQP is right-angled at Q; therefore PQ ⟂ OQ.