The diagram shows a circle, centre the origin - OCR - GCSE Maths - Question 17 - 2017 - Paper 1
Question 17
The diagram shows a circle, centre the origin.
(a) Write down the equation of the circle.
(b) Point P has coordinates (8, –6).
Show that point P lies on the circle... show full transcript
Worked Solution & Example Answer:The diagram shows a circle, centre the origin - OCR - GCSE Maths - Question 17 - 2017 - Paper 1
Step 1
Write down the equation of the circle.
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Answer
The equation of a circle with center at the origin (0, 0) and radius r is given by:
x2+y2=r2
From the diagram, we can see the radius is 10. Thus, the equation of the circle is:
x2+y2=100
Step 2
Show that point P lies on the circle.
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Answer
To determine if point P (8, -6) lies on the circle, we will substitute x = 8 and y = -6 into the circle's equation:
Calculate:
82+(−6)2=64+36=100
Since 100=100, point P lies on the circle.
Step 3
Find the equation of the tangent to the circle at point P.
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Answer
The gradient of the radius to point P can be calculated using the coordinates of the origin (0, 0) and point P (8, -6):
Calculate the gradient of the radius:
m_{radius} = rac{y_2 - y_1}{x_2 - x_1} = rac{-6 - 0}{8 - 0} = - rac{3}{4}
The gradient of the tangent line is the negative reciprocal of the radius's gradient:
m_{tangent} = rac{4}{3}
Using the point-slope form of the equation of a line, the equation of the tangent line at point P (8, -6) is:
y - (-6) = rac{4}{3}(x - 8)
