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Step-by-Step Solution
Step 1: Determine the center and radius of circle C₁
The circle C₁ 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 C₁.
Since the diameter is 4, the radius of C₁ is 2.
Step 2: Equation of circle C₁
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 C₁
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 C₁, 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 C₁.
Step 4: Defining circle C₂ with OA as diameter
We now consider a new circle C₂ whose diameter is the segment OA of the first circle.
By definition, the tangent to C₂ 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 C₂ and intercepts on axes
The tangent at A for circle C₂ 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 $.
