Two Dimensional Analytical Geometry-II: One Mark Questions, Book Back Answers and Notes

About Two Dimensional Analytical Geometry-II

Hard ~180 min study

Analytical geometry, which goes back to René Descartes, describes curves by equations in $x$ and $y$. Once a curve has an equation, questions about its shape, its highest point or where a line touches it become algebra problems. In Class 11 you did this for straight lines. This chapter does it for circles and the conic sections.

The conics are the curves you get when a plane cuts a double cone. The chapter uses a single definition for them: a conic is the locus of a point $P$ that moves so that its distance from a fixed point (the focus) is $e$ times its distance from a fixed line (the directrix). The constant $e$ is the eccentricity. If $e = 1$ the curve is a parabola, if $0 < e < 1$ it is an ellipse, and if $e > 1$ it is a hyperbola. One idea covers three curves, and the standard equations all follow from it.

For the exam, you need the standard forms at your fingertips, together with the formulas for foci, vertices, directrices and length of latus rectum. Many problems start with an equation that is not in standard form, so completing the square is a skill you will use again and again. Tangents and normals, the condition for a line to be a tangent, and word problems on parabolic arches and bridges also appear. The ideas are used again in physics for projectile paths and planetary orbits.

What you'll learn

• Write the equation of a circle from given conditions and find the centre and radius from the general form.
• Identify the conic represented by a second-degree equation using the coefficients of $x^2$ and $y^2$.
• Find the vertex, centre, foci, directrices, eccentricity and length of latus rectum of a given conic.
• Find the equations of the tangent and normal at a point on a circle or conic.
• Use parametric forms of the parabola, ellipse and hyperbola in problems.
• Solve word problems on parabolic arches, suspension bridges and elliptical paths.

Before you start

• The Cartesian plane, the distance formula and the section formula.
• Completing the square in $x$ and in $y$.
• Slope of a line and the different forms of the equation of a straight line.

Two Dimensional Analytical Geometry-II explained

Circles and Conics in Coordinate Geometry

Circles

A circle is the locus of points at a fixed distance $r$ from a fixed centre. With centre $(h, k)$ the equation is $(x-h)^2 + (y-k)^2 = r^2$. The general form $x^2 + y^2 + 2gx + 2fy + c = 0$ has centre $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$. If $g^2 + f^2 - c = 0$ the circle reduces to a single point (a point circle), and if $g^2 + f^2 - c < 0$ there is no real circle. The chapter also finds circles through given points, circles on a given diameter, and tangents and normals to a circle. The line $y = mx + c$ touches $x^2 + y^2 = a^2$ when $c^2 = a^2(1 + m^2)$.

Conics and Eccentricity

A conic is the locus of a point whose distance from the focus divided by its distance from the directrix is the constant $e$. The value of $e$ decides the curve: $e = 1$ parabola, $e < 1$ ellipse, $e > 1$ hyperbola. To identify a conic from $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ with $B = 0$, the textbook uses these tests: $A = C$ gives a circle; $A = 0$ or $C = 0$ (but not both) gives a parabola; $A \neq C$ with $A$ and $C$ of the same sign gives an ellipse; $A$ and $C$ of opposite signs gives a hyperbola. Some special cases give only a point or no real locus.

Parabola

Every point of a parabola is equally far from the focus and the directrix. The standard form $y^2 = 4ax$ ($a > 0$) has vertex $(0, 0)$, focus $(a, 0)$, directrix $x = -a$, axis $y = 0$ and latus rectum of length $4a$; it opens to the right. The other three forms are $y^2 = -4ax$ (left), $x^2 = 4ay$ (up) and $x^2 = -4ay$ (down). A shifted parabola such as $(y-k)^2 = 4a(x-h)$ has its vertex at $(h, k)$, and every other feature moves with it. A focal chord is any chord through the focus; the latus rectum is the focal chord perpendicular to the axis.

Ellipse and Hyperbola

For an ellipse, the sum of the distances of any point from the two foci is constant and equals $2a$. The standard form is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$, foci $(\pm ae, 0)$, directrices $x = \pm\frac{a}{e}$ and $b^2 = a^2(1 - e^2)$. The major axis has length $2a$ and the minor axis $2b$. For a hyperbola, the difference of the distances from the foci is constant, equal to $2a$. Its standard form is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with foci $(\pm ae, 0)$, directrices $x = \pm\frac{a}{e}$, $b^2 = a^2(e^2 - 1)$, transverse axis $2a$ and conjugate axis $2b$. A hyperbola also has two asymptotes, $y = \pm\frac{b}{a}x$, which the branches approach but never meet. The length of the latus rectum is $\frac{2b^2}{a}$ for both curves.

Parametric Forms, Tangents and Normals

In parametric form each point is written using one variable: $(at^2, 2at)$ on $y^2 = 4ax$, $(a\cos\theta, b\sin\theta)$ on the ellipse, and $(a\sec\theta, b\tan\theta)$ on the hyperbola. Parameters make tangent and normal equations shorter to write. For example, the tangent to $y^2 = 4ax$ at $(at^2, 2at)$ is $yt = x + at^2$. The line $y = mx + c$ is a tangent to $y^2 = 4ax$ if $c = \frac{a}{m}$, to the ellipse if $c^2 = a^2m^2 + b^2$, and to the hyperbola if $c^2 = a^2m^2 - b^2$.

Real-Life Applications

The last section applies the chapter to real situations. Planets move in ellipses with the Sun at one focus, as Kepler found. A satellite dish or a torch reflector is parabolic, so rays parallel to the axis meet at the focus, or light from the focus leaves parallel. Cooling towers have hyperbolic cross-sections. Textbook problems include finding heights on a parabolic arch or on the cable of a suspension bridge, and working with elliptical arches.

Common mistakes to avoid

• Mixing up ellipse and hyperbola: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is an ellipse, while a minus sign between the terms gives a hyperbola. Similarly $b^2 = a^2(1 - e^2)$ belongs to the ellipse and $b^2 = a^2(e^2 - 1)$ to the hyperbola.
• Finding the vertex or centre of a shifted conic correctly but then giving the foci and directrices as if the centre were at the origin.
• Taking the major axis along the x-axis when the larger denominator is under $y^2$. In $\frac{x^2}{9} + \frac{y^2}{25} = 1$ the major axis lies along the y-axis.
• Using the wrong tangency condition, for example $c^2 = a^2m^2 + b^2$ (ellipse) for a hyperbola, where it should be $c^2 = a^2m^2 - b^2$.
• Forgetting that a circle can be seen as an ellipse with $a = b$, which gives eccentricity 0.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

What is the difference between the standard and general forms of a circle?

The standard form $(x-h)^2 + (y-k)^2 = r^2$ shows the centre $(h, k)$ and radius $r$ directly. The general form $x^2 + y^2 + 2gx + 2fy + c = 0$ is the expanded version. You either complete the square or use centre $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$. Both describe the same circle.

How do I know if a second-degree equation represents a parabola?

For $Ax^2 + Cy^2 + Dx + Ey + F = 0$ (no $xy$ term), it is a parabola when exactly one of $x$ and $y$ is squared, that is, $A = 0$ or $C = 0$ but not both. For example, $y^2 - 8x + 2y + 17 = 0$ is a parabola. Completing the square in $y$ gives $(y+1)^2 = 8(x-2)$, with vertex $(2, -1)$.

Why are parametric equations useful?

They describe every point on the curve with a single variable, $t$ or $\theta$. That makes it easy to name a general point, write its tangent and normal, and handle questions about chords and loci. For instance, $(a\cos\theta, b\sin\theta)$ covers the whole ellipse as $\theta$ runs from $0$ to $2\pi$.

What does eccentricity tell us about a conic?

It measures how far the curve is from being a circle. A circle has $e = 0$. An ellipse has $0 &lt; e &lt; 1$ and gets flatter as $e$ approaches 1. A parabola has $e = 1$. A hyperbola has $e &gt; 1$, and its branches open wider as $e$ increases.

Where are conics used in real life?

Parabolic reflectors in satellite dishes, telescopes and vehicle headlights focus signals or light. An elliptical reflector sends waves from one focus to the other, which is how lithotripsy machines break kidney stones. Cooling towers use hyperbolic shapes, and planets move in elliptical orbits around the Sun.

Does a hyperbola have a centre like an ellipse?

Yes. The centre is the midpoint of the two vertices, and also of the two foci. The curve does not pass through the centre, but the two asymptotes cross there. For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ the centre is the origin.