A circle has equation $x^2 + y^2 = 12.25$ - Edexcel - GCSE Maths - Question 24 - 2022 - Paper 2
Question 24
A circle has equation $x^2 + y^2 = 12.25$.
The point P lies on the circle.
The coordinates of P are (2.1, 2.8).
The line L is the tangent to the circle at point P.... show full transcript
Worked Solution & Example Answer:A circle has equation $x^2 + y^2 = 12.25$ - Edexcel - GCSE Maths - Question 24 - 2022 - Paper 2
Step 1
Find the center and radius of the circle
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Answer
The equation of the circle is given by x2+y2=12.25. This is in the standard form (x−h)2+(y−k)2=r2, where (h,k) is the center and r is the radius.
From the equation, we see that the center is at (0,0) and the radius is:
r=extsqrt(12.25)=3.5.
Step 2
Check that point P lies on the circle
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Answer
Substitute the coordinates of point P, (2.1, 2.8), into the equation of the circle:
2.12+2.82=12.25
Calculating gives:
4.41+7.84=12.25,
which confirms that point P lies on the circle.
Step 3
Find the slope of the radius at point P
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Answer
The slope of the radius from the center (0,0) to point P (2.1, 2.8) is given by:
ext{slope} = rac{y_2 - y_1}{x_2 - x_1} = rac{2.8 - 0}{2.1 - 0} = rac{2.8}{2.1}.
To simplify, divide both by 0.1:
ext{slope} = rac{28}{21} = rac{4}{3}.
Step 4
Determine the slope of the tangent line L
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Answer
The slope of line L, the tangent, is the negative reciprocal of the slope of the radius:
ext{slope of L} = - rac{1}{ ext{slope of radius}} = - rac{3}{4}.
Step 5
Use point-slope form to find the equation of L
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Answer
Using point-slope form y−y1=m(x−x1) with point P (2.1, 2.8) and slope - rac{3}{4}:
y - 2.8 = - rac{3}{4}(x - 2.1).
Distributing gives:
y - 2.8 = - rac{3}{4}x + rac{6.3}{4},
which simplifies to:
y = - rac{3}{4}x + 1.575 + 2.8.
Calculating 2.8+1.575=4.375,
leads us to:
y = - rac{3}{4}x + 4.375.
Step 6
Convert to standard form ax + by = c
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Answer
To convert to standard form:
Multiply through by 4 to eliminate the fraction:
4y=−3x+17.5.
Rearranging gives:
3x+4y=17.5.
Since a, b, and c must be integers, multiply through by 2 to clear the decimal:
6x+8y=35.
Thus, the equation of line L is:
6x+8y=35.
