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The discriminant is a valuable tool in identifying the type of conic section represented by a quadratic equation such as \[ Ax^2 + Bxy + Cy^2 = 0 \].For conic sections, the discriminant is given by the formula \( D = B^2 - 4AC \). This simple calculation provides insight into whether the conic is a parabola, ellipse, or hyperbola:

• If \(D = 0\), the conic is a parabola.
• If \(D > 0\), the conic is a hyperbola.
• If \(D < 0\), the conic is an ellipse.

In our exercise, the equation provided is \[ 153x^2 + 192xy + 97y^2 = 225 \]. Upon computing, we find \( D = 192^2 - 4 \times 153 \times 97 = -22500 \), which is less than zero. This indicates that the given equation represents an ellipse. The discriminant helps to determine the immediate nature of the conic without graphically plotting the equation, simplifying the process considerably.Breaking down the calculation and its implications helps students quickly identify conic sections and comprehend the fundamental geometry that results.

The rotation of axes is utilized when a conic equation has an \(xy\) term, making it difficult to identify and graph easily. This method aids in simplifying the equation so that the \(xy\)-term is removed, and the conic's inherent properties are more evident.To eliminate the \(xy\)-term, a specific angle \(\theta\) is used based on the formula \[ \tan(2\theta) = \frac{B}{A - C} \].Using this equation in our example, we calculate \[ \tan(2\theta) = \frac{192}{153 - 97} = \frac{192}{56} \approx 3.4286 \].To find \(\theta\), we calculate \[ 2\theta = \tan^{-1}(3.4286) \approx 73.74^{\circ} \],giving us \(\theta \approx 36.87^{\circ}\).With this angle, we apply a transformation using the rotation matrix, expressing the original coordinates \((x, y)\) as new coordinates \((x', y')\):- \( x = x'\cos(\theta) - y'\sin(\theta) \)- \( y = x'\sin(\theta) + y'\cos(\theta) \)By substituting these transformed coordinates back into the original equation, the \(xy\)-term vanishes, confirming the simplification. This process not only reveals the conic’s simplified form but also maintains its orientation relative to the original axes.

An ellipse is a closed curve on a plane surrounding two focal points. The sum of the distances from any point on the ellipse to the two foci is constant. When the equation has no \(xy\)-term, identifying ellipse properties becomes straightforward.The standard form of an ellipse centered at the origin can be given as:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Key characteristics of an ellipse include:

• The longest diameter (major axis) runs through both foci.
• The shortest diameter (minor axis) is perpendicular to the major axis.
• If \(a > b\), the major axis is along the \(x\)-direction; if \(b > a\), it's along the \(y\)-direction.

With our rotated axes, the initial equation transforms, showcasing the conic’s characteristics clearly. The absence of the \(xy\)-term after rotation allows us to identify these axes, essential for accurately sketching the ellipse correctly.By understanding ellipse geometry, students can sketch and measure any ellipse confidently, leveraging these consistent properties to analyze angular orientation, size, and symmetry.