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CH-20-001 · Coordinate circle, tangent intercepts and triangle area

-10-8-6-4-20246810-10-8-6-4-2246810xyP

The circle x2+y2=73x^2+y^2=73 has centre O. P(3, 8) lies on the circle. The tangent at P meets the positive x-axis at A and the positive y-axis at B. Find the equation of the tangent and the exact area of triangle OAB. Show all your working.

Equation of Tangent to a Circle & Triangle Area Booklet p. 37 & p. 84
Method
1) Find the radius gradient
2) Take its negative reciprocal for the tangent gradient
3) Use point-gradient form
4) Find both intercepts
5) Use ½ × base × height
Worked Example (CH-20-001)
1)
Gradient of radius OPGradient of tangentEquation through P
8−03−0=83\frac{8-0}{3-0}=\frac{8}{3}−1÷83=−38-1\div\frac{8}{3}=-\frac{3}{8}y−8=−38(x−3)y-8=-\frac{3}{8}(x-3)
2)
Simplify: 3x+8y=733x+8y=73.
3)
On the x-axis, y=0: A=(733,0)A=(\frac{73}{3},0). On the y-axis, x=0: B=(0,738)B=(0,\frac{73}{8}).
4)
The axes are perpendicular, so Area(OAB)=12×733×738=532948\text{Area}(OAB)=\frac12\times\frac{73}{3}\times\frac{73}{8}=\frac{5329}{48} square units.
Ans = 3x+8y=733x+8y=73; area 532948\frac{5329}{48} square units.