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Chapter 10: Problem 20

(a) \(f(x, y)=0\) is the equation of a circle. (b) \(f(x, y) \equiv x^{2}+2 y^{2}-3 x-y+2\).

Short Answer

Expert verified
No, the given function does not represent a circle.

Step by step solution

01

Identify the General Form

The general form of the equation of a circle is \[Ax^2 + Ay^2 + Bx + Cy + D = 0\].Set the given function equal to zero: \[f(x, y) = x^2 + 2y^2 - 3x - y + 2 = 0\]. In this form, a circle requires the coefficients of \(x^2\) and \(y^2\) to be equal.
02

Compare Coefficients

Compare the given function \(f(x, y) = x^2 + 2y^2 - 3x - y + 2\) to the general form. Here, the coefficients of \(x^2\) and \(y^2\) are 1 and 2 respectively. Notice that they are not equal.
03

Conclusion

Since the coefficients of \(x^2\) and \(y^2\) are not equal (1 鈮� 2), the given equation does not represent a circle.

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Key Concepts

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

general form of a circle
Understanding the general form of a circle's equation is the first step in recognizing and working with circles in algebra. The equation of a circle in general form is written as \[Ax^2 + Ay^2 + Bx + Cy + D = 0\].Here, * \(A\), \(B\), \(C\), and \(D\) are coefficients
* \(x\) and \(y\) are the variables representing coordinates in the plane
The crucial aspect of a circle's general equation is that the coefficients of \(x^2\) and \(y^2\) must be equal. This ensures that the equation maintains the standard structure of a circle. If these coefficients are not equal, the resulting shape will not be a perfect circle.

To see this in context, consider the given function: \[f(x, y) = x^2 + 2y^2 - 3x - y + 2 = 0\]. This step sets us up to compare coefficients and determine if the function indeed describes a circle.
coefficients
Coefficients in algebra are the numerical or constant part attached to variables in expressions. In the general form equation of a circle, the coefficients play a pivotal role in defining the shape. For \[Ax^2 + Ay^2 + Bx + Cy + D = 0\], we focus on the values represented by \(A\), \(B\), \(C\), and \(D\).
* \(A\) affects how \(x^2\) and \(y^2\) scale
* \(B\) and \(C\) influence the linear terms for \(x\) and \(y\)
* \(D\) represents the constant term in the equation
In our equation, \[f(x, y) = x^2 + 2y^2 - 3x - y + 2 = 0\], we can identify * The coefficient of \(x^2\) as 1
* The coefficient of \(y^2\) as 2
* The coefficient of \(x\) as -3
* The coefficient of \(y\) as -1
* Constant term as 2

These coefficients will help us understand the geometric properties of the shape that's represented by the equation.
comparison of coefficients
Comparing coefficients is a crucial step in determining whether a given equation fits the criteria for being a circle. For an equation to represent a circle, the coefficients of \(x^2\) and \(y^2\) must be identical. Based on the general form \[Ax^2 + Ay^2 + Bx + Cy + D = 0\], we compare the given function \[f(x, y) = x^2 + 2y^2 - 3x - y + 2 = 0\].
We observe:
* Coefficient of \(x^2\) is 1
* Coefficient of \(y^2\) is 2
Clearly, 1 is not equal to 2. This inconsistency indicates that the given equation does not fulfill the criteria for a circle.

Understanding this aspect helps in quickly identifying the nature of the equation by just comparing the coefficients of \(x^2\) and \(y^2\). If they match, it's a good indication you're dealing with a circle; if not, it's something else.

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Most popular questions from this chapter

A line with gradient 2 which passes through the point \((k, 0)\) will touch the parabola \(y^{2}=4 a x\) if and only if : (a) \(k>0\) (b) \(k<0\) (c) \(k=-a\) (d) \(k<-a\) (e) \(4 k=-a\).

Find the equation of the tangent to the curve \(a y^{2}=x^{3}\) at the point \(\left(a t^{2}, a t^{3}\right)\), where \(a>0\) and \(t\) is a parameter. (U of L)

Show that the point with coordinates \((2+2 \cos \theta, 2 \sin \theta)\) lies on the circle \(x^{2}+y^{2}=4 x, \quad\) and obtain the equation of the tangent to the circle at this point. The tangents at the points \(\mathrm{P}\) and \(\mathrm{Q}\) on this circle touch the circle \(x^{2}+y^{2}=1\) at the points \(\mathrm{R}\) and \(\mathrm{S}\). Find the coordinates of the point of intersection of these tangents, and obtain the equation of the circle through the points \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) and \(\mathrm{S}\). (U of \(\mathrm{L})\)

The parametric equations of a curve are \(x=\sec \theta+1, \quad y=\tan \theta-1\). Its Cartesian equation is: (a) \(y^{2}+3=x^{2}\) (b) \(x^{2}-y^{2}-2 x-2 y=1\) (c) \(x^{2}-y^{2}+2 x+2 y+1=0\) (d) \(x^{2}-y^{2}=1\) (e) \((x-1)^{2}=y^{2}\).

Show that the equation of the normal to the parabola \(y^{2}=4 a x\) at the point \(\left(a t^{2}, 2 a t\right)\) is \(y+t x=2 a t+a t^{3}\). If this normal meets the parabola again at the point \(\left(a T^{2}, 2 a T\right)\) show that $$ t^{2}+t T+2=0 $$ and deduce that \(T^{2}\) cannot be less that 8 . The line \(3 y=2 x+4 a\) meets the parabola at the points \(P\) and \(Q\). Show that the normals at \(\mathrm{P}\) and \(\mathrm{Q}\) meet on the parabola.
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