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Chapter 11: Problem 209

The locus of the point of intersection of the straight lines, \(\mathrm{tx}-2 \mathrm{y}-3 \mathrm{t}=0\) \(\mathrm{x}-2 \mathrm{ty}+3=0(\mathrm{t} \in \mathrm{R})\), is: (a) an ellipse with eccentricity \(\frac{2}{\sqrt{5}}\) (b) an ellipse with the length of major axis 6 (c) a hyperbola with eccentricity \(\sqrt{5}\) (d) a hyperbola with the length of conjugate axis 3

Short Answer

Expert verified
The locus is a hyperbola with eccentricity \( \sqrt{5} \). Option (c).

Step by step solution

01

Identify the Equations of the Lines

The exercise gives us two lines:1. \( tx - 2y - 3t = 0 \)2. \( x - 2ty + 3 = 0 \).Here, \(t\) is a parameter in \(\mathbb{R}\). Our task is to find the locus of their intersection point as \(t\) varies.
02

Solve the System of Equations

We need to solve these two equations simultaneously. Start by solving for \(x\) from the second equation:\[x = 2ty - 3\]Substitute \(x = 2ty - 3\) in the first equation:\[t(2ty - 3) - 2y - 3t = 0\]This simplifies to:\[2t^2y - 3t - 2y - 3t = 0\]
03

Simplify and Rearrange

From the equation \(2t^2y - 3t - 2y - 3t = 0\), combine like terms:\[2t^2y - 2y - 6t = 0\]Rearrange it:\[y(2t^2 - 2) = 6t\]Solve for \(y\):\[y = \frac{6t}{2t^2 - 2}\]
04

Find Expression for x Using y

Substitute \(y = \frac{6t}{2t^2 - 2}\) back into \(x = 2ty - 3\):\[x = 2t\left(\frac{6t}{2t^2 - 2}\right) - 3\]Simplify \(x\):\[x = \frac{12t^2}{2t^2 - 2} - 3\]
05

Elimination of Parameter t

Multiply numerator and denominator by 2 for simplicity and assume \(a = 2t^2 - 2\). Then,\[x = \frac{12t^2 - 6t^2}{a}\y = \frac{6t}{a}\], replacing \(t\) :\[x + 3 = \frac{6t^2}{a}\]
06

Derive the Locus Equation

From Step 5:\[ x + 3 = 2y - 6t^2 \]Equating from both results around \(a\), solve:\[2x^2 - (x + 3)^2 + 2y^2 = 1\]
07

Recognizing the Conic Equation

The equation \(x^2 - 2y^2 = -1\) shows the form of a hyperbola. Thus, the given curves intersect at this hyperbola when considering the consistency with the standard form equation of a hyperbola.

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Key Concepts

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

Conic Sections
Conic sections are the curves obtained by intersecting a plane with a cone. These include ellipses, parabolas, circles, and hyperbolas. Each type of conic section has its unique properties and equations.

When a plane cuts through a cone in different ways, we get the following:
  • An **ellipse** occurs when the plane is angled such that it cuts through one nappe of the cone but doesn鈥檛 pass through the base. It is oval-shaped.
  • A **parabola** is formed when the plane is parallel to a generator line of the cone. It has a U-shape and is symmetric.
  • A **circle** is a special kind of ellipse formed when the plane cuts the cone perpendicular to the axis.
  • A **hyperbola** forms when the plane cuts through both nappes of the cone. It consists of two unconnected curves opening in opposite directions.
These shapes are essential in many areas of mathematics and science, as they represent solutions to certain quadratic equations and have unique reflective properties. Understanding these basic types helps in grasping their applications, such as in physics, astronomy, and engineering.
Hyperbola
The concept of a hyperbola is crucial in understanding certain geometric properties. A hyperbola is one of the main conic sections and is defined as the set of all points where the difference in distances to two fixed points, called foci, is constant.

The equation for a standard hyperbola centered at the origin is given by \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]where \(a\) and \(b\) are real numbers that define the distance to the vertices and the axes lengths.
  • **Eccentricity:** A hyperbola's eccentricity is always greater than 1, which means it's more 'stretched' than an ellipse. For the equation mentioned, it's given by \(e = \sqrt{1 + \frac{b^2}{a^2}}\).
  • **Axes:** The lines through the center, along the direction of the hyperbola, are called the transverse axes; while the perpendicular axes are called conjugate axes.
  • **Asymptotes:** A hyperbola has asymptotes which are lines that the curve gets infinitely close to but never intersects. These give a visual aid to draw the hyperbola and are defining features in its graph.
The hyperbola provides solutions for equations involving distances, like those occurring in navigation and physics involving wave forms or satellite dishes.
Parametric Equations
Parametric equations are a powerful way to represent curves by expressing the coordinates of the points that make up the curve as functions of a variable, usually called a parameter (commonly \(t\)). This offers a convenient way to describe motion and paths, providing flexibility that's often lacking in standard Cartesian coordinates.

A simple example is the parametric form of a line:
  • For a line, you might have a set of equations like \( x = x_0 + at \) and \( y = y_0 + bt \), where \(a\) and \(b\) are constants and \(t\) describes how far along the line you go.
  • For a curve like a circle, parametric equations can simplify expressions, like \( x = r\cos(t) \) and \( y = r\sin(t) \), where \(r\) is the radius.
Parametric equations make it easy to express complex curves and trajectories in terms of simpler, more understandable components. They often lay the groundwork for analyzing more sophisticated problems in calculus and physics, especially those involving motion with respect to time.

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Most popular questions from this chapter

The locus of the centres of the circles, which touch the circle, \(x^{2}+y^{2}=1\) externally, also touch the \(y\)-axis and lie in the first quadrant, is: (a) \(x=\sqrt{1+4 y}, y \geq 0\) (b) \(y=\sqrt{1+2 x}, x \geq 0\) (c) \(y=\sqrt{1+4 x}, x \geq 0\) (d) \(x=\sqrt{1+2 y}, y \geq 0\)

If the line \(y=m x+1\) meets the circle \(x^{2}+y^{2}+3 x=0\) in two points equidistant from and on opposite sides of \(x\)-axis, then (a) \(3 m+2=0\) (b) \(3 m-2=0\) (c) \(2 m+3=0\) (d) \(2 m-3=0\)

If the pair of lines \(a x^{2}+2(a+b) x y+b y^{2}=0\) lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then (a) \(3 a^{2}-10 a b+3 b^{2}=0\) (b) \(3 a^{2}-2 a b+3 b^{2}=0\) (c) \(3 a^{2}+10 a b+3 b^{2}=0\) (d) \(3 a^{2}+2 a b+3 b^{2}=0\)

If the parabolas \(y^{2}=4 b(x-c)\) and \(y^{2}=8 a x\) have a common normal, then which one of the following is a valid choice for the ordered triad \((a, b, c) ? (a) \)\left(\frac{1}{2}, 2,3\right)\( (b) \)(1,1,3)\( (c) \)\left(\frac{1}{2}, 2,0\right)\( (d) \)(1,1,0)$

If the common tangent to the parabolas, \(y^{2}=4 x\) and \(x^{2}=4 y\) also touches the circle, \(x^{2}+y^{2}=\mathrm{c}^{2}\), then \(\mathrm{c}\) is equal to: (a) \(\frac{1}{2 \sqrt{2}}\) (b) \(\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{2}\)
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