8. [tex]x^2y - x^2 + 2xy - 2x + 3 = 0[/tex]
9. [tex]x^3 - x^2y - xy^2 + y^3 + 2x^2 - 4y^2 + 2xy + x + y + 1 = 0[/tex]
10. [tex]x^3 + 2x^2y - xy^2 - 2y^3 + xy - y^2 - 10 = 0[/tex]
11. [tex]x^3 + 2x^2y - xy^2 - 2y^3 + 3xy - 3y^2 + 4x + 7 = 0[/tex]
12. [tex]6x^3 - 11x^2y + 6xy^2 - y^3 + 2x + 3y + 7 = 0[/tex]
13. [tex]4x^3 - x^2y - 4xy^2 + y^3 + 3x^2 + 2xy - y^2 + x + y + 7 = 0[/tex]
14. [tex]x^3 + x^2y - xy^2 - y^3 - 3x + 5 = 0[/tex]
15. [tex]x^3 + x^2y - xy^2 - y^3 + x^2 - y^2 + 5 = 0[/tex]
16. [tex]x^3 + y^3 - 3ax^2 = 0[/tex]
17. [tex](x - y + 1)(x - y - 2)(x + y) = 8x - 1[/tex]
18. Show that the eight points of intersection of the curve [tex]xy(x^2 - y^2) + x^2 + y^2 = a^2[/tex] and its asymptotes lie on a circle whose centre is at the origin.
19. Find the equation to the cubic which has the same asymptotes as the curve [tex]x^3 - 6x^2y + 11xy^2 - 6y^2 - x + y + 1 = 0[/tex] and which passes through the points [tex](0, 0), (1, 1)[/tex] and [tex](0, 1)[/tex].
20. Find the asymptotes of the curve [tex](y - x)(y - 2x)^2 + (y + 3x)(y - 2x) + 2x + 2y - 1 = 0[/tex] and show that their points of intersection with the curve lie on a straight line.
21. Find the asymptotes of the curves.
(i) [tex]r \cos \theta = a \sin \theta[/tex]
(ii) [tex]r = a \log \theta[/tex]
(iii) [tex]r \sin \theta = a \cos 2\theta[/tex]
(iv) [tex]r \cos \theta = a \sin^2 \theta[/tex]
(v) [tex]r \theta = ae^{\theta}[/tex]
(vi) [tex]r \sin \theta = e^{\theta}[/tex]
(vii) [tex]r = a \sec \theta[/tex]
Answer (1)
The question involves understanding various properties of curves like finding asymptotes, proving certain geometric properties about points of intersection, and dealing with polar coordinates and parametric forms. Here's a breakdown of approaches for some of these tasks:
Equation of the cubic with the same asymptotes
To find the equation of a cubic curve with the same asymptotes as another curve, analyze the given curve's terms to determine its behavior at infinity. For the curve x 3 − 6 x 2 y + 11 x y 2 − 6 y 2 − x + y + 1 = 0 , look closely at the terms x 3 , x 2 y , x y 2 as these dominate the behavior when x or y become very large in magnitude. The cubic that shares these asymptotes will have similar terms.
Asymptotes of another curve and their intersections with the curve
For curves like ( y − x ) ( y − 2 x ) 2 + ( y + 3 x ) ( y − 2 x ) + 2 x + 2 y − 1 = 0 , the asymptotes can be found by focusing on terms where the degree in x and y matches the degree of the constant term when set equal to zero. Figuring out how to equate the higher degree terms can help find the asymptotes' equations. Once known, calculate their intersection points with the original curve, and you often show geometric properties like collinearity using these points.
Polar curves and asymptotes
Finding asymptotes in polar equations like r cos θ = a sin θ involves setting the function in a form where r approaches infinity or the angles approach significant values like θ → 0 , π /2 . This is similar for other expressions given in polar, logarithmic, or exponential forms.
Each step involves transforming the equation to find asymptotes and intersections, paying attention to limits, and behavior at infinity. These are complex tasks ideally tackled step-by-step and with reference to a textbook or by consulting a detailed focus on polynomial and transcendental curve properties.
