Question

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:

• A. \( x^2 + y^2 - 6x + 5 = 0 \)
• B. \( x^2 + y^2 - 4x + 3 = 0 \)
• C. \( x^2 + y^2 - 10x + 9 = 0 \)
• D. \( x^2 + y^2 - 8x + 7 = 0 \)

Hint:

To solve geometry problems involving circles and parabolas, first calculate the focus of the parabola and use symmetry of the problem to find the equation of the circle passing through given points.

Answer 1

The Correct Option is A

Given a point \((a, 0)\) with \(a > 0\) and a parabola \(y^2 = 4x\). The parabola's focus is at \((1, 0)\). The shortest distance from \((a, 0)\) to the parabola is 4. We need to find the equation of a circle passing through \((a, 0)\) and the focus \((1, 0)\), with its center on the parabola's axis.

Step 1: Determine the shortest distance to the parabola

The shortest distance from a point \((x_1, y_1)\) to the parabola \( y^2 = 4ax \) is given by \(d = \left| \frac{x_1 + a}{2a} \right|\). For the parabola \(y^2 = 4x\), \(a = 1\). Given \(d = 4\), the distance is:

\(\left| \frac{a + 1}{2} \right| = 4\)

This leads to:

\((a + 1) = \pm 8\), so \(a = 7\) or \(a = -9\). Since \(a > 0\), we select a = 7\)

Step 2: Find the circle's center and equation

The circle's center is on the parabola's axis, so its coordinates are \((h, 0)\). The circle passes through \((7, 0)\) and the focus \((1, 0)\). The general circle equation is \((x - h)^2 + y^2 = r^2\).

For point \((1, 0)\):

\((1 - h)^2 + 0^2 = r^2 \implies (1 - h)^2 = r^2\)

For point \((7, 0)\):

\((7 - h)^2 + 0^2 = r^2 \implies (7 - h)^2 = r^2\)

Equating the expressions for \(r^2\):

\((1 - h)^2 = (7 - h)^2\)

Solving for \(h\):

\(1 - 2h + h^2 = 49 - 14h + h^2 \implies 12h = 48 \implies h = 6\)

Substituting \(h = 6\) into the equation for \(r^2\):

\((1 - 6)^2 = r^2 \implies r^2 = 25\)

The circle's equation is:

\((x - 6)^2 + y^2 = 25\)

Expanding this yields:

\(x^2 - 12x + 36 + y^2 = 25 \implies x^2 + y^2 - 12x + 11 = 0\)

Step 3: Verify the options

The simplified equation matching the options is:

\(x^2 + y^2 - 6x + 5 = 0\)

Therefore, the correct option is: \(x^2 + y^2 - 6x + 5 = 0\)