The circle shown in the diagram has, as tangents, the x-axis, the y-axis, the line $x + y = 2$ and the line $x + y = 2k$, where $k > 1$ - Leaving Cert Mathematics - Question 3 - 2012
Question 3
The circle shown in the diagram has, as tangents, the x-axis, the y-axis, the line $x + y = 2$ and the line $x + y = 2k$, where $k > 1$.
Find the value of $k$.
Worked Solution & Example Answer:The circle shown in the diagram has, as tangents, the x-axis, the y-axis, the line $x + y = 2$ and the line $x + y = 2k$, where $k > 1$ - Leaving Cert Mathematics - Question 3 - 2012
Step 1
Find the radius and center of the circle
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Answer
The radius (r) is the distance from the center to any of the tangents. Considering the tangents as lines, we can outline the relationship:
Since the circle is tangent to the x-axis and y-axis, the center (r,r) must be equidistant from both axes. Therefore, we can say the radius is equal to r.
The line x+y=2 can be represented as:
r2=(r+k)2+(r+k)2
This leads us to:
r = 2 + rac{ ext{sqrt}(2)}{2}
Step 2
Use the midpoint formula
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Answer
The midpoint (r,r) is the average of the endpoints: (1,1) and (k,k), thus leading to:
k+1=2rk=2r−1
Consequently, we need to express k in terms of r.
Step 3
Solve for k
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Answer
We have the equation:
2−extsqr(2)=4
Implying that:
k=3+2extsqrt(2)
To ensure k>1, we need to check that:
k=3+2extsqrt(2)>1 holds true, which it does.
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