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Step-by-Step Solution
Step 1: Determine the center and radius of circle Cā
The circle Cā passes through the origin O(0,0) and has diameter 4 on the positive x-axis.
So, the endpoints of this diameter are at (0,0) and (4,0).
The midpoint of these points is (2,0), which is the center of Cā.
Since the diameter is 4, the radius of Cā is 2.
Step 2: Equation of circle Cā
A circle with center (2,0) and radius 2 has the standard form
$ (x - 2)^2 + y^2 = 4 $.
Expanding:
$ (x - 2)^2 + y^2 = 4 $
becomes
$ x^2 - 4x + 4 + y^2 = 4 $,
which simplifies to
$ x^2 + y^2 - 4x = 0 $.
Step 3: Identify chord OA on circle Cā
The line containing the chord OA is given by
$ y = 2x $.
This line clearly passes through O(0,0) (the origin).
When intersecting the circle Cā, it will pass through two points: one is O(0,0) and the other is A (the second intersection point).
Thus, OA is a chord of Cā.
Step 4: Defining circle Cā with OA as diameter
We now consider a new circle Cā whose diameter is the segment OA of the first circle.
By definition, the tangent to Cā at A will be perpendicular to its diameter OA.
Step 5: Slope relationships
The line OA has slope
$ m_{\text{OA}} = 2$
(from the equation $y = 2x$).
Any line perpendicular to it will have slope
$ m = -\frac{1}{2} $.
Step 6: Tangent at A to circle Cā and intercepts on axes
The tangent at A for circle Cā is perpendicular to OA, so it has slope
$ -\frac{1}{2} $
(as mentioned above).
This tangent meets the x-axis at P and the y-axis at Q.
We want to find the ratio
$ QA : AP $.
Step 7: Calculating the ratio Q A : A P
A key property is that the tangent at A to the circle with diameter OA is perpendicular to OA.
Using coordinate geometry or similar triangles, one can deduce that
$ \frac{QA}{AP} = \frac{1}{\tan^2\theta} $
if the slope of OA is
$ \tan\theta $.
Since $ \tan\theta = 2 $ for OA, then
$ \tan^2\theta = 4 $.
Therefore,
$ \displaystyle \frac{QA}{AP} = \frac{1}{4} $,
which gives the ratio
$ QA : AP = 1 : 4 $.
Hence, the required ratio is
$ \displaystyle 1 : 4 $.
