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Chapter 20: Problem 63

\(\mathrm{PN}\) 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

Short Answer

Expert verified
(D) none of these.

Step by step solution

01

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.
02

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 \(N(0, y_1)\).
03

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: \[ Q \left( \frac{a^2 x_1 - b^2 a}{a^2 + b^2}, \frac{a^2 y_1}{a^2 + b^2} \right) \]
04

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.
05

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.
06

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

An ellipse has \(O B\) as semi minor axis, \(F\) and \(F^{\prime}\) its focii and the angle \(F B F^{\prime}\) is a right angle. Then the eccentricity of the ellipse is (A) \(\frac{1}{\sqrt{2}}\) (B) \(\frac{1}{2}\) (C) \(\frac{1}{4}\) (D) \(\frac{1}{\sqrt{3}}\)

The number of point(s) outside the hyperbola \(\frac{x^{2}}{25}-\frac{y^{2}}{36}=1\) from where two perpendicular tangents can be drawn to the hyperbola is/are (A) none (B) 1 (C) 2 (D) infinte

The tangent at the point ' \(\alpha\) ' on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) meets the auxiliary circle in two points which subtend a right angle at the centre. The eccentricity of the ellipse is (A) \(\frac{1}{\sqrt{1+\sin ^{2} \alpha}}\) (B) \(\frac{1}{\sqrt{1+\cos ^{2} \alpha}}\) (C) \(\sqrt{1+\sin ^{2} \alpha}\) (D) none of these

The centroid of the triangle formed by the feet of the normals from the point \((h, k)\) to the parabola \(y^{2}+4 a x\) \(=0,(a>0)\) lies on (A) \(x\)-axis (B) \(y\)-axis (C) \(x=h\) (B) \(y=k\)

The maximum area of an isosceles triangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with its vertex at one end of the major axis is (A) \(\sqrt{3} a b\) (B) \(\frac{3 \sqrt{3}}{4} a b\)(C) \(\frac{5 \sqrt{3}}{4} a b\) (D) none of these
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