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Magazine Chapter 11: Problem 222
The locus of a point \(P(\alpha, \beta)\) moving under the condition that the line \(y=\alpha x+\beta\) is a tangent to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is (a) an ellipse (b) a circle (c) a parabola (d) a hyperbola
Short Answer
Expert verified
(d) a hyperbola
Step by step solution
01
Equation of the tangent line
The condition given is that the line \(y = \alpha x + \beta\) is tangent to the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\). The equation of any tangent to the hyperbola can be written as \(y = mx \pm \sqrt{a^2m^2 - b^2}\) where \(m\) is the slope of the tangent line.
02
Match line equation to tangent formula
We know that the line \(y = \alpha x + \beta\) must match the form of the tangent line equation. This implies that \(m = \alpha\) and \(\beta = \pm \sqrt{a^2\alpha^2 - b^2}\).
03
Simplifying the condition
Square both sides of the equation \(\beta = \pm \sqrt{a^2\alpha^2 - b^2}\) to remove the square root: \(\beta^2 = a^2\alpha^2 - b^2\). Rearranging gives \(\beta^2 = a^2\alpha^2 - b^2\).
04
Rewrite in standard form
Rewrite the equation \(\beta^2 = a^2\alpha^2 - b^2\) as \(a^2\alpha^2 - \beta^2 = b^2\). Divide by \(b^2a^2\) to obtain: \(\frac{\alpha^2}{\frac{b^2}{a^2}} - \frac{\beta^2}{b^2} = 1\).
05
Identify the type of conic section
The standard form obtained \(\frac{\alpha^2}{\frac{b^2}{a^2}} - \frac{\beta^2}{b^2} = 1\) matches the general form of a hyperbola \(\frac{x^2}{p^2} - \frac{y^2}{q^2} = 1\). Therefore, the locus of point \(P(\alpha, \beta)\) is a hyperbola.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hyperbola
A hyperbola is a fascinating type of conic section characterized by its open curve formed by intersecting a plane with a double cone. Unlike ellipses and circles, hyperbolas consist of two disjoint, mirror-image branches. These branches are symmetrically placed relative to the center of the hyperbola.
- A hyperbola is defined by its two focus points, and any point on the hyperbola has a constant difference in distance to these focus points.
- The general equation for a hyperbola oriented along the x-axis is: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). This tells us the relationship between the x and y coordinates for any point on the curve.
- In a hyperbola's equation, if the x terms are subtracted from the y terms, the hyperbola opens along the y-axis instead.
Tangent Line
A tangent line to a hyperbola or any curve is a line that intersects the curve at exactly one point. This unique intersection point is called the "point of tangency." With hyperbolas, tangent lines play a crucial role in understanding the curve鈥檚 geometry.
- A tangent line only touches a hyperbola at one point, implying at that point, the line has the exact same slope as the curve.
- The equation of the tangent line to a hyperbola can also demonstrate some symmetry, as it involves the slope of the tangent touching the curve and the structure of the hyperbola鈥檚 equation.
- Formulating and solving tangent line equations is critical in many applications, such as finding specific points or verifying if a given line is tangent to the hyperbola.
Conic Sections
Conic sections are shapes derived from slicing a cone at different angles and positions. They include circles, ellipses, parabolas, and hyperbolas. Each of these shapes has unique equations and properties that define their geometric form.
- Circular and elliptical sections are formed through slicing a cone parallel to its base and at an angle less steep than the side of the cone, respectively.
- If cut parallel to one of the sides, the result is a parabola, which has only one point of focus.
- Finally, intersecting the cone with a plane arriving at both nappes creates a hyperbola, showcasing its distinct double-branch structure.
Equation of Tangent
The equation of a tangent is key in identifying if a line is tangent to a conic section such as a hyperbola. This equation changes based on the conic it's applied to.
- For a hyperbola defined by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equation of the tangent line becomes \(y = mx \pm \sqrt{a^2m^2 - b^2}\), where \(m\) is the slope.
- This specific form helps in ensuring that the line only touches at one point, maintaining tangency.
- Equations like these are instrumental not just in theoretical problems but also in practical scenarios, like designing tangent paths in engineering structures.
