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RV-24-001 · Coordinate tangent and distance from the centre

A circle has equation x2+y2=32x^2+y^2=32. P(4, 4) and Q(20, -12) are points in the coordinate plane. O is the origin.
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(a) Show that the line PQ is tangent to the circle at P.

(b) Calculate OQ and PQ, to 2 decimal places.

Part (a) Coordinate Circle Tangent Proof & Length of a Line Booklet p. 80, p. 82 & p. 84
Method
1) Check x² + y² = r²
2) Show m_OP × m_PQ = −1 (radius ⊥ tangent)
3) Use distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²) for OQ and PQ
Worked Example (RV-24-001 · Part (a))
Show that the line PQ is tangent to the circle at P.
1)
4² + 4² = 32, so P is on the circle.
2)
Gradient OP = 11; gradient PQ = −1-1.
3)
Their product is −1. A line perpendicular to a radius at its endpoint is a tangent.
Ans = P lies on the circle, and OP is perpendicular to PQ, so PQ is tangent at P.
Part (b) Coordinate Circle Tangent Proof & Length of a Line Booklet p. 80, p. 82 & p. 84
Method
1) Check x² + y² = r²
2) Show m_OP × m_PQ = −1 (radius ⊥ tangent)
3) Use distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²) for OQ and PQ
Worked Example (RV-24-001 · Part (b))
Calculate OQ and PQ, to 2 decimal places.
1)
Use the distance formula for OQ.
2)
23.32
3)
Use the distance formula for PQ (or Pythagoras in OPQ).
4)
22.63
Ans = OQ = 23.32; PQ = 22.63.