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GB-42-05-001 · Equation of a tangent

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A circle has equation x2+y2=85x^2+y^2=85. Find the equation of the tangent at P(9, 2). Give your answer in the form y = mx + c.

Equation of Tangent to a Circle Booklet p. 81 & p. 84
Method
1) Find the radius gradient
2) Take its negative reciprocal
3) Substitute into point-gradient form
4) Clear the denominator, expand and rearrange
Worked Example (GB-42-05-001)
1)
Find the gradient of the radius from (0,0)(0,0) to P(9,2)P(9,2): mradius=2−09−0=29m_{\mathrm{radius}}=\frac{2-0}{9-0}=\frac{2}{9}.
2)
The tangent is perpendicular to the radius: mtangent=−1÷29=−92m_{\mathrm{tangent}}=-1\div\frac{2}{9}=-\frac{9}{2}.
3)
Use y−y1=m(x−x1)y-y_1=m(x-x_1): y−2=−92(x−9)y-2=-\frac{9}{2}(x-9).
4)
Clear the denominator before expanding: 2(y−2)=−9(x−9)2(y-2)=-9(x-9).
5)
Expand and rearrange: 2y−4=−9x+81⇒y=−92x+8522y-4=-9x+81\Rightarrow y=-\frac{9}{2}x+\frac{85}{2}.
Ans = y=−92x+852y=-\frac{9}{2}x+\frac{85}{2}