Problem 63 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\mathrm{PN}\) is the ordinate of any point \(P\) on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(A A^{\prime}\) is its transverse axis. If \(Q\) divides \(A P\) in the ratio \(a^{2}: b^{2}\), then \(N Q\) is (A) \(\perp\) to \(A^{\prime} P\) (B) parallel to \(A^{\prime} P\) (C) \(\perp\) to \(O P\) (D) none of these

Step-by-Step Solution

Answer

(D) none of these.

Step 1: Understand the Problem

We need to determine the relationship between line segment \(NQ\) and other geometrical elements of the hyperbola given the hyperbola equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). We have a point \(P\) and point \(Q\) that divides \(AP\) in the ratio \(a^2: b^2\). We aim to find whether \(NQ\) is perpendicular or parallel to specified lines.

Step 2: Analyze Hyperbola Setup

The hyperbola centers at origin \(O(0,0)\), transverse axis along \(x\)-axis with endpoints \(A(-a,0)\) and \(A'(a,0)\). Point \(P(x_1, y_1)\) lies on hyperbola such that it satisfies \(\frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} = 1\). \(N\) is on the \(y\)-axis, meaning \(N(0, y_1)\).

Step 3: Calculate Coordinates of Q

Let \(A(-a, 0)\) and \(P(x_1, y_1)\). \(Q\) divides \(AP\) in ratio \(a^2 : b^2\), real ratio is \(\frac{a^2}{a^2 + b^2}\) and \(\frac{b^2}{a^2 + b^2}\). Therefore, coordinates for \(Q\) are given by: \[ Q \left( \frac{a^2 x_1 + b^2(-a)}{a^2 + b^2}, \frac{a^2 y_1 + b^2 \cdot 0}{a^2 + b^2} \right) \] These simplify to: \[ Q \left( \frac{a^2 x_1 - b^2 a}{a^2 + b^2}, \frac{a^2 y_1}{a^2 + b^2} \right) \]

Step 4: Determine the Slope of NP and QP

For \(NP\), slope is: \[ \text{slope of } NP = \frac{y_1 - y_1}{x_1 - 0} = 0 \] For \(QP\), slope is: \[ \text{slope of } QP = \frac{y_1 - \frac{a^2 y_1}{a^2 + b^2}}{x_1 - \frac{a^2 x_1 - b^2 a}{a^2 + b^2}} \] Simplifying:\[ \text{slope of } QP = \frac{b^2 y_1}{(a^2 + b^2)x_1 - (a^2 x_1 - b^2 a)} \] This simplifies to zero if analysis is kept consistent with geometry.

Step 5: Analyze Relationship Between NQ and AP

The calculated coordinates don't produce basic perpendicularity pattern like horizontal-vertical, thus \(NQ\) isn't simply perpendicular to \(AP\) or other necessary segments directly by y-axis alignments or other basic geometric properties.

Step 6: Evaluate Perpendicularity of NQ to A'P

\(AP\) involves direction along \(x\)-axis initially, revisiting possible perpendicular analysis involves challenging hyperbola settings, coordinate placement continually suggests directly conducing perpendicular components not calculated with holistic adjustments toward \(A'P\). Perform checks mechanics otherwise non-responsive directly from given conditions.

Key Concepts

Ordinate of Hyperbola

In the geometry of a hyperbola, the ordinate refers to the vertical coordinate or the "y"-value of any point on the curve. Given the hyperbola equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), each point \(P(x_1, y_1)\) not only lies on the curve but must also satisfy this equation.

The ordinate \(PN\), associated with point \(P(x_1, y_1)\), is essentially the distance from \(P\) to the x-axis, where \(N\) lies directly below or above \(P\) along the y-axis. This value \(y_1\) plays a crucial role in defining the exact position of point \(P\) with respect to the center of the hyperbola, also known as the origin \(O(0, 0)\).

• The ordinate \(PN\) is visualized as a line parallel to the transverse axis intersecting \(P\).
• In hyperbola graphing, the ordinate is key to plotting points that reflect the shape and orientation of the hyperbola.

Transverse Axis of Hyperbola

The transverse axis of a hyperbola is the line segment that passes through the two vertices of the hyperbola and is the primary axis along which the hyperbola opens.

In our given equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the transverse axis aligns with the x-axis. The endpoints are the vertices of the hyperbola at points \(A(-a, 0)\) and \(A'(a, 0)\). Thus, the length of the transverse axis is \(2a\).

• The transverse axis helps determine the direction in which the hyperbola opens. Here, it opens horizontally along the x-direction.
• Such an axis is essential when considering the properties of the hyperbola, such as its vertices, foci, and later, its asymptotes.
• In sketching or solving problems, recognizing the transverse axis provides insight into the hyperbola's orientation and structure.

Ratio Division in Geometry

Ratio division in geometry refers to a method of dividing a line segment into two parts based on specified ratio values. It helps in finding a precise point \(Q\) such that it divides \(AP\) in the ratio \(a^2:b^2\).

For our problem, if point \(A\) is \((-a, 0)\) and point \(P\) is \((x_1, y_1)\), point \(Q\) divides \(AP\) in this specified ratio. The coordinates of \(Q\) are computed using the section formula as follows:

\[ Q \left( \frac{a^2 x_1 + b^2(-a)}{a^2 + b^2}, \frac{a^2 y_1 + b^2 \cdot 0}{a^2 + b^2} \right) \]
This ratio division is pivotal in various branches of geometry to position a point accurately along a line segment based on desired proportional distances.

• The formula used here effectively weighs the coordinates according to the given ratios to determine the precise division point.
• Understanding the concept of ratio division helps in tackling a variety of geometric constructions and proofs.
• This method is frequently employed in problems involving complex geometric loci and configurations.