91Ó°ÊÓ

Trigonometric identities are fundamental formulas in trigonometry used to simplify expressions and solve equations. They establish relationships between the trigonometric functions such as sine, cosine, and tangent. In this exercise, the identities were crucial to verify if points lie on a hyperbola, specifically the identity:

• \( \sec^2 \theta - \tan^2 \theta = 1 \)

This identity helps confirm that the point \( P(3\sec\theta, 2\tan\theta) \) is on the hyperbola. Here, \( \sec \theta \) is the reciprocal of \( \cos \theta \) and \( \tan \theta \) is the ratio of \( \sin \theta \) to \( \cos \theta \). Likewise, the same identity assures that \( Q(3\sec\phi, 2\tan\phi) \) also lies on the hyperbola when \( \theta + \phi = \frac{\pi}{2} \). Therefore, understanding and manipulating such trigonometric identities underpins the verification of these points and the resulting analyses.

Normals are lines perpendicular to a tangent line at a point on a curve. For the hyperbola, the equation of the normal at any point \((x_0,y_0)\) is given by:

• \( \frac{xx_0}{9} - \frac{yy_0}{4} = \frac{x^2_0}{9} - \frac{y^2_0}{4} \)

This equation is derived by taking derivatives of the hyperbola equation, representing the slopes of the tangents. Normals are important in determining intersection points, like in this exercise where we find the intersection of normals at two distinct points, \( P \) and \( Q \). Substituting the coordinates of point P or Q allows us to write specific normal equations:

• For \( P(3\sec\theta, 2\tan\theta) \): \[ x \sec \theta - \frac{y \tan \theta}{2} = 1 \]
• For \( Q(3\sec\phi, 2\tan\phi) \):\[ x \sec \phi - \frac{y \tan \phi}{2} = 1 \]

These normals eventually intersect, and their intersection plays a critical role in further calculations, giving insight into the behavior of the hyperbola and associated lines.

Simultaneous equations are sets of equations with multiple variables, requiring solutions that satisfy all equations simultaneously. In this exercise, the normals' equations, both derived from distinct points on the hyperbola, form simultaneous equations:

• \( x \sec \theta - \frac{y \tan \theta}{2} = 1 \)
• \( x \sec \phi - \frac{y \tan \phi}{2} = 1 \)

To solve these, we use given trigonometric relationships: since \(\theta + \phi = \frac{\pi}{2}\), use \( \sec \phi = \csc \theta \) and \( \tan \phi = \cot \theta \). By substituting these into the equations, you can isolate variables, typically focusing on \( y \) since that's often the value you're solving for in hyperbola problems. Solving this adjusted pair provides the ordinate of the intersections of the normals, which in this case ultimately evaluates to \( y = -\frac{13}{2} \). Recognizing how simultaneous equations help in narrowing down to specific variable values forms the backbone of solving multi-variable mathematical problems like this one.