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

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: \(x^2 + y^2 - 6x + 5 = 0\)

Answer 2

The equation of the given parabola is: \[ y^2 = 4x. \] This is of the form \( y^2 = 4ax \), where \( a = 1 \). Therefore, the focus of the parabola is at \( (1, 0) \). We are given that the shortest distance from the point \( (a, 0) \) to the parabola is 4. This condition helps in determining the position of the center of the circle. Next, the equation of the circle passing through the point \( (a, 0) \) and the focus \( (1, 0) \) and having its center on the axis of the parabola is given by: \[ (x - h)^2 + y^2 = r^2, \] where \( h \) is the center's x-coordinate and \( r \) is the radius. By solving using the distances from the center to the points \( (a, 0) \) and \( (1, 0) \), we find that the equation of the circle is: \[ x^2 + y^2 - 6x + 5 = 0. \]