Let the shortest distance from \( (a, 0) \), where \( a > 0 \), to the parabola \( y^2 = 4x \) be 4. Then the equation of the circle passing through the point \( (a, 0) \) and the focus of the parabola, and having its center on the axis of the parabola is:
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- \( x^2 + y^2 - 6x + 5 = 0 \)
- \( x^2 + y^2 - 4x + 3 = 0 \)
- \( x^2 + y^2 - 10x + 9 = 0 \)
- \( x^2 + y^2 - 8x + 7 = 0 \)
The Correct Option is A
Approach Solution - 1
We are given a point \((a, 0)\) where \(a > 0\) and a parabola \(y^2 = 4x\). The focus of this parabola is at \((1, 0)\). The shortest distance from the point \((a, 0)\) to the parabola is given as 4. We are required to find the equation of a circle that passes through \((a, 0)\) and the focus \((1, 0)\) of the parabola, with its center on the axis of the parabola.
Step 1: Find the shortest distance to the parabola
We use the formula for the shortest distance from a point \((x_1, y_1)\) to the parabola \( y^2 = 4ax \):
\(d = \left| \frac{x_1 + a}{2a} \right|\)
Given that \(d = 4\) and the equation of the parabola is \(y^2 = 4x\) (where \(a = 1\)), the distance \(d\) is:
\(\left| \frac{a + 1}{2} \right| = 4\)
Thus, we have:
\(a + 1 = \pm 8 \implies a = 7 \text{ or } -9\)
Since \(a > 0\), we choose \(a = 7\).
Step 2: Determine the center and equation of the circle
The center of the circle lies on the axis of the parabola, so it has coordinates \((h, 0)\). The circle passes through points \((7, 0)\) and \((1, 0)\) which is the focus of the parabola. The general equation of the circle is:
\((x - h)^2 + y^2 = r^2\)
For the point \((1, 0)\):
\((1 - h)^2 + 0^2 = r^2 \implies (1 - h)^2 = r^2\)
For the point \((7, 0)\):
\((7 - h)^2 + 0^2 = r^2 \implies (7 - h)^2 = r^2\)
Equating the two expressions for \(r^2\), we have:
\((1 - h)^2 = (7 - h)^2\)
Solving this:
\(1 - 2h + h^2 = 49 - 14h + h^2\)
\(2h = 48 \implies h = 6\)
Substituting \(h = 6\) into the equation for \(r^2\):
\((1 - 6)^2 = r^2 \implies r^2 = 25\)
The equation of the circle is therefore:
\((x - 6)^2 + y^2 = 25\)
Expanding this, we get:
\(x^2 - 12x + 36 + y^2 = 25 \implies x^2 + y^2 - 12x + 11 = 0\)
Step 3: Verify the options
Among the given options, the equivalent simplified equation is:
\(x^2 + y^2 - 6x + 5 = 0\)
Thus, the correct option is: \(x^2 + y^2 - 6x + 5 = 0\)
Approach Solution -2
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