Question

Match the conics in Column I with the statements/expressions in Column II. (A) Circle (p) The locus of the point \((h, k)\) for which the line (B) Parabola \(h x+k y=1\) touches the circle \(x^{2}+y^{2}=4\) (C) Ellipse (q) Points \(z\) in the complex plane satisfying (D) Hyperbola \(\quad|z+2|-|z-2|=\pm 3\) (r) Points of the conic have parametric representation \(x=\sqrt{3}\left(\frac{1-t^{2}}{1+t^{2}}\right), y=\frac{2 t}{1+t^{2}}\) (s) The eccentricity of the conic lies in the interval \(1 \leq x<\infty\) (t) Points \(z\) in the complex plane satisfying \(\operatorname{Re}(z+1)^{2}=|z|^{2}+1\)

Short answer

(A) matches with (p), since the line is tangent to the circle. (B) matches with (r), providing a parametric form indicating a parabola. (C) matches with (q), due to the constant difference from two foci characteristic of an ellipse. (D) matches with (s) and (t), as they describe the conditions and eccentricity of a hyperbola.

Step-by-step solution

Identify the Circle

Analyze the given line and how it relates to the circle. A tangent to a circle is a line that touches the circle at exactly one point. The provided equation, \(hx + ky = 1\), represents a family of lines. The condition that this line is tangent to the circle \(x^2 + y^2 = 4\) can be solved using the tangent-line relationship for circles. A line \(lx + my = n\) is tangent to \(x^2 + y^2 = r^2\) if \(r^2(l^2+m^2)=n^2\), so for this case, we want to find if there exists a pair \(h, k\) such that \(4(h^2+k^2)=1\).

Identify the Parabola

Seek an expression or statement that represents a standard form of a parabola. Since the given equation \(hx + ky = 1\) potentially relates to a parabola, we need to recall the definition of a parabola, which is the set of points equidistant from a point, called the focus, and a line called the directrix. However, none of the statements describes such a relationship directly, so we look for a differentiating feature unique to parabolas. Statement (r) provides a parametric representation where \(x\) and \(y\) can be expressed in terms of a parameter \(t\).

Identify the Ellipse

Consider the expression that suggests an ellipse. In complex numbers, the statement \(q: |z+2|-|z-2|=\pm 3\) implies a constant difference between distances to two fixed points, the foci. This condition defines an ellipse in the complex plane with foci at \(z = -2\) and \(z = 2\), where the distance between any point on the ellipse and the foci differs by exactly 3.

Identify the Hyperbola

Look for the statement or expression that best describes a hyperbola. Hyperbolas are characterized by an eccentricity that is greater than 1, which is represented by the statement \(s: 1 \leq x < \infty\). Moreover, the complex representation \(t: \operatorname{Re}(z + 1)^2 = |z|^2 + 1\) is indicative of a hyperbola's equation in the complex plane.

Match Statements with Conics

With all the identifications made, match each conic section with its corresponding statement or expression. (A) - Circle matches with \(p\), since a tangent line to the circle at a point \(h, k\) is described by the linear equation given in \(p\). (B) - Parabola matches with \(r\), because it is represented by the parametric equation given. (C) - Ellipse matches with \(q\) due to the condition of constant difference from two foci, described in \(q\). (D) - Hyperbola matches with \(s\) and \(t\), because they both describe conditions of a hyperbola.

Key concepts

Circle Tangent

Understanding the concept of a circle tangent is fundamental when dealing with conic sections. A tangent to a circle is a straight line that touches the circle at exactly one point. This single point of contact is known as the point of tangency.

Now, let's dive into how to identify a circle tangent. Given a circle's equation, such as \(x^2 + y^2 = r^2\), and a line expressed as \(lx + my = n\), a specific condition must be met for the line to be tangent to the circle: the perpendicular drop from the center of the circle to the line should be equal to the radius of the circle. In algebraic terms, this condition translates to \(r^2(l^2 + m^2) = n^2\).

Now, applying this concept to the provided exercise, the equation \(hx + ky = 1\) represents a family of lines. For one of these to be tangent to the circle \(x^2 + y^2 = 4\), we check if there is a pair (h, k) that satisfies the tangency condition. If such a pair exists, it aligns with the given circle, ensuring the line touches the circle at just one point.

Ellipse Complex Plane

Ellipses can be represented in the complex plane, offering a unique and powerful way to explore their properties using complex numbers. A point \(z\) in the complex plane satisfies an ellipse equation if the sum or difference of its distances (known as focal radii) from two fixed points (foci of the ellipse) is a constant.

Imagine an ellipse with foci located at \(z = -2\) and \(z = 2\). If we take any point on the ellipse, represented by the complex number \(z\), the mathematical condition is given by \(|z + 2| - |z - 2| = \text{constant}\). This relationship perfectly describes the nature of an ellipse in the complex plane, resulting from the intersection of a plane with a double-napped cone.

In our exercise, the given constant is \(\text{±3}\), defining the specific ellipse in question. Therefore, the equation \(|z+2|-|z-2|= \text{±3}\) implies that for all points \(z\) on the ellipse, the sum or difference of distances to the foci is a constant value (in this case, 3), which is a distinctive property of ellipses.

Hyperbola Eccentricity

The eccentricity of a conic section is a measure that describes its deviation from being circular. A hyperbola, one of the conic sections, always has an eccentricity greater than one. This value, denoted usually by \(e\), differentiates hyperbolas from other conic sections such as circles (eccentricity = 0), ellipses (eccentricity < 1), and parabolas (eccentricity = 1).

Eccentricity of a hyperbola can be understood by the ratio of the distance from any point on the hyperbola to a focus of the hyperbola, to the perpendicular distance from that point to the nearest directrix. As the eccentricity increases, the hyperbola's opening becomes more 'stretched'. In the context of the exercise, the statement that eccentricity lies in the interval \(1 \< x \textless \text{∞}\) clearly identifies the conic section as a hyperbola.

Further, a hyperbola's relationship in the complex plane can often be exploited to analyze its properties. For instance, the complex equation \(\text{Re}(z+1)^2=|z|^2+1\) relates to a hyperbola because it implies the squared real part of the complex number once offset by one has a certain relationship to the magnitude of the complex number—an expression reminiscent of a hyperbola's Cartesian equation.

Parabola Parametric Representation

A parabola is a curve where every point is at an equal distance from a fixed point (focus) and a straight line (directrix). The parametric representation of a parabola involves describing the parabola's points using a parameter, commonly denoted as \(t\). This approach allows us to express the Cartesian coordinates \(x\) and \(y\) as functions of \(t\).

For instance, the parametric form \(x = \text{f}(t)\) and \(y = \text{g}(t)\) could define any curve, and for a parabola, these functions have a specific relation to the parameter. In the exercise solution, the equations \(x=\text{√3}\bigg(\frac{1-t^2}{1+t^2}\bigg)\) and \(y=\frac{2t}{1+t^2}\) represent the parabola in parametric form.

These expressions ensure that for each real value of \(t\), there is a corresponding point on the parabola, and together they trace the entire curve. The relation also reveals the symmetry and directrix of the parabola. Parametric equations are very useful in calculus and analytical geometry because they allow easy differentiation and integration of the curve's points.