Finding the Circle's Intercept on the Y-Axis

The question asks us to find the length of the intercept cut by the given circle on the y-axis. The equation of the circle is \(x^2 + y^2 + 4x – 7y + 12 = 0\).

Understanding the Y-Intercept of a Circle

The points where a circle intersects the y-axis are the points where the x-coordinate is zero. To find these points, we substitute \(x=0\) into the circle's equation.

Step-by-Step Calculation

Let's substitute \(x=0\) into the given equation:

\(0^2 + y^2 + 4(0) – 7y + 12 = 0\)

This simplifies to a quadratic equation in terms of \(y\):

\(y^2 – 7y + 12 = 0\)

To find the y-coordinates where the circle crosses the y-axis, we need to solve this quadratic equation. We can factor the quadratic expression:

We look for two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4.

So, we can write the equation as:

\((y – 3)(y – 4) = 0\)

Setting each factor equal to zero gives us the possible values for \(y\):

\(y – 3 = 0 \implies y = 3\)

\(y – 4 = 0 \implies y = 4\)

The circle intersects the y-axis at two points: \((0, 3)\) and \((0, 4)\).

Calculating the Intercept Length

The length of the intercept on the y-axis is the distance between these two points of intersection. Since both points lie on the y-axis, the distance is simply the absolute difference of their y-coordinates.

Intercept length \(= |y_2 – y_1| = |4 – 3| = 1\).

Alternative Method: Using the Formula

The general equation of a circle is \(x^2 + y^2 + 2gx + 2fy + c = 0\). The length of the intercept on the y-axis is given by the formula \(2\sqrt{f^2 – c}\), provided that \(f^2 – c \ge 0\) (for real intersection points).

Comparing the given equation \(x^2 + y^2 + 4x – 7y + 12 = 0\) with the general equation, we have:

\(2g = 4 \implies g = 2\)

\(2f = –7 \implies f = –7/2\)

\(c = 12\)

Now, let's calculate \(f^2 – c\):

\(f^2 – c = (-\frac{7}{2})^2 – 12 = \frac{49}{4} – 12 = \frac{49 – 48}{4} = \frac{1}{4}\)

Since \(f^2 – c = \frac{1}{4} > 0\), the circle intersects the y-axis at two distinct points.

Using the formula for the y-intercept length:

Length \(= 2\sqrt{f^2 – c} = 2\sqrt{\frac{1}{4}} = 2 \times \frac{1}{2} = 1\)

Both methods yield the same result.

The length of the intercept cut by the circle on the y-axis is 1.

Revision Table: Circle Intercept Concepts

Additional Information: Conditions for Intercepts

For a circle with equation \(x^2 + y^2 + 2gx + 2fy + c = 0\):

The circle cuts the y-axis at two distinct points if \(f^2 > c\).

The circle touches the y-axis (one intercept point) if \(f^2 = c\).

The circle does not intersect the y-axis if \(f^2 < c\).

Similarly, for the x-intercepts:

The circle cuts the x-axis at two distinct points if \(g^2 > c\).

The circle touches the x-axis if \(g^2 = c\).

The circle does not intersect the x-axis if \(g^2 < c\).

In our case, \(f = -7/2\) and \(c = 12\). \(f^2 = (-7/2)^2 = 49/4 = 12.25\). Since \(f^2 (12.25) > c (12)\), the circle cuts the y-axis at two distinct points, and an intercept exists.