Circles C₁ and C₂ have equations
$(x + 5)^2 + (y - 6)^2 = 9$
and $x^2 + y^2 - 6x - 16 = 0$ respectively.
(a) Write down the centres and radii of C₁ and C₂.
(b) S... show full transcript
Worked Solution & Example Answer:Circles C₁ and C₂ have equations
$(x + 5)^2 + (y - 6)^2 = 9$
and $x^2 + y^2 - 6x - 16 = 0$ respectively - Scottish Highers Maths - Question 4 - 2016
Step 1
Write down the centres and radii of C₁ and C₂.
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Answer
To find the centre and radius of circle C₁ from the equation
(x+5)2+(y−6)2=9, we can use the standard form of a circle's equation:
(x−h)2+(y−k)2=r2
where (h,k) is the center and r is the radius.
From the given equation, we can identify:
Centre of C₁: (−5,6)
Radius of C₁: r1=extsqrt(9)=3.
For circle C₂, we simplify the equation x2+y2−6x−16=0:
Rearranging the equation gives:
x2−6x+y2−16=0
Completing the square for the x terms:
(x−3)2−9+y2−16=0
This simplifies to:
(x−3)2+y2=25
Thus, we can identify:
Centre of C₂: (3,0)
Radius of C₂: r2=extsqrt(25)=5.
Step 2
Show that C₁ and C₂ do not intersect.
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Answer
To determine whether the two circles intersect, we first calculate the distance between their centers:
The distance d between the centers (−5,6) and (3,0) can be found using the distance formula:
d=extsqrt((x2−x1)2+(y2−y1)2)
Substituting in the coordinates gives:
d=extsqrt((3−(−5))2+(0−6)2)=extsqrt((3+5)2+(−6)2)=extsqrt(82+62)=extsqrt(64+36)=extsqrt(100)=10.
Next, we compare this distance with the sum of the radii:
Sum of the radii: r1+r2=3+5=8.
Since d=10>8=r1+r2, we conclude that the circles C₁ and C₂ do not intersect.
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