Two Dimensional Analytical Geometry-II — Study Notes
Two Dimensional Analytical Geometry-II — Study Notes
This chapter studies the four conics — the circle, parabola, ellipse and hyperbola — through their Cartesian equations. The single idea that unifies them is the focus–directrix law: every conic is the path traced by a point whose distance from a fixed point keeps a fixed ratio to its distance from a fixed line. These notes summarise the standard forms, the geometry behind each formula, and the moves that examiners reward.
1. The circle
A circle is the set of points kept at a constant distance (the radius) from a fixed centre. With centre \((h,k)\) and radius \(r\), squaring the distance condition gives the standard form
\[(x-h)^{2}+(y-k)^{2}=r^{2}.\]Multiplying out and renaming the constants produces the general form \(x^{2}+y^{2}+2gx+2fy+c=0\), whose centre is \((-g,-f)\) and radius \(\sqrt{g^{2}+f^{2}-c}\). The expression under the root decides the nature of the locus:
- \(g^{2}+f^{2}-c>0\): a genuine (real) circle;
- \(g^{2}+f^{2}-c=0\): a single point (a point circle);
- \(g^{2}+f^{2}-c<0\): no real points at all.
Two shortcuts appear again and again in problems. If \((x_{1},y_{1})\) and \((x_{2},y_{2})\) are the ends of a diameter, the circle is \((x-x_{1})(x-x_{2})+(y-y_{1})(y-y_{2})=0\), because any third point on the circle sees the diameter at a right angle. And given a circle \(S=0\) and a line \(L=0\), the equation \(S+\lambda L=0\) sweeps out every circle through the two points where the line cuts \(S\); choosing \(\lambda\) lets you pin down the particular circle you need (one tangent to an axis, one through a given point, and so on).
To locate a point \((x_{1},y_{1})\) relative to a circle, evaluate \(S_{1}=x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c\): the point is outside, on, or inside according as \(S_{1}\) is positive, zero, or negative. When the point is outside, \(\sqrt{S_{1}}\) is the length of the tangent drawn from it.
Tangent and normal to a circle
The tangent at a point \((x_{1},y_{1})\) of \(x^{2}+y^{2}=a^{2}\) is obtained by the “split the squares” rule, \(xx_{1}+yy_{1}=a^{2}\); for a general circle it reads \(xx_{1}+yy_{1}+g(x+x_{1})+f(y+y_{1})+c=0\). The normal is the line through the point and the centre, so for \(x^{2}+y^{2}=a^{2}\) it passes through the origin. A line \(y=mx+c\) just grazes \(x^{2}+y^{2}=a^{2}\) when the centre-to-line distance equals \(a\); algebraically this is \(c^{2}=a^{2}(1+m^{2})\), and the tangent lines are \(y=mx\pm a\sqrt{1+m^{2}}\).
2. Conics and eccentricity
Fix a point \(S\) (the focus), a line \(\ell\) (the directrix) and a positive number \(e\) (the eccentricity). A point \(P\) lies on the conic when
\[\frac{SP}{PM}=e,\]where \(PM\) is the perpendicular distance from \(P\) to \(\ell\). The value of \(e\) names the curve: \(e=1\) gives a parabola, \(0
3. Parabola
Because \(e=1\), a parabola is equidistant from its focus and directrix. Placing the vertex at the origin gives four standard equations. For \(y^{2}=4ax\) (opening right) the focus is \((a,0)\), the directrix is \(x=-a\), the axis is the \(x\)-axis, and the latus rectum — the focal chord perpendicular to the axis — has length \(4a\). Replacing \(x\) by \(-x\) opens it left (\(y^{2}=-4ax\)); swapping the roles of \(x\) and \(y\) opens it up or down (\(x^{2}=4ay\) and \(x^{2}=-4ay\)). When the vertex moves to \((h,k)\), shift each variable accordingly.
Key vocabulary: the axis is the line of symmetry through the focus; the vertex is where the axis meets the curve; a focal chord is any chord through the focus; the latus rectum is the focal chord at right angles to the axis.
4. Ellipse
For \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) with \(a>b\), the major axis lies along the \(x\)-axis with length \(2a\), the minor axis has length \(2b\), and the foci sit at \((\pm c,0)\) where \(c^{2}=a^{2}-b^{2}\). The eccentricity is \(e=\dfrac{c}{a}<1\) and the directrices are \(x=\pm\dfrac{a}{e}\). The latus rectum has length \(\dfrac{2b^{2}}{a}\). If instead \(b>a\) (written with the larger denominator under \(y^{2}\)), the major axis is vertical and the foci move onto the \(y\)-axis — always read off which denominator is larger before quoting foci.
The defining feature used constantly in problems is that the sum of the two focal distances of any point on the ellipse equals the major-axis length \(2a\). The auxiliary circle, drawn on the major axis as diameter (\(x^{2}+y^{2}=a^{2}\)), is the bridge to the parametric description of the ellipse.
5. Hyperbola
For \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\) the transverse axis has length \(2a\), the conjugate axis \(2b\), the foci are \((\pm c,0)\) with \(c^{2}=a^{2}+b^{2}\), and \(e=\dfrac{c}{a}>1\). The directrices are \(x=\pm\dfrac{a}{e}\) and the latus rectum is \(\dfrac{2b^{2}}{a}\). The hyperbola is the only one of the four conics with asymptotes: the lines \(y=\pm\dfrac{b}{a}x\), which the branches approach but never touch. Its signature property is that the absolute difference of the two focal distances is constant and equals \(2a\). Writing the equation with the larger term under \(y^{2}\) (a \(-\) sign in front of the \(x^{2}\) term) turns it into a vertical hyperbola with foci on the \(y\)-axis.
6. Parametric forms
A parameter lets a single variable trace the whole curve, which is convenient for tangents and for “a point on the conic” arguments.
- Circle \(x^{2}+y^{2}=a^{2}\): \((a\cos\theta,\,a\sin\theta)\).
- Parabola \(y^{2}=4ax\): \((at^{2},\,2at)\).
- Ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\): \((a\cos\theta,\,b\sin\theta)\), where \(\theta\) is the eccentric angle.
- Hyperbola \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\): \((a\sec\theta,\,b\tan\theta)\).
7. Tangents and normals to conics
For all four conics the tangent at a point is found by the same “split the squares and products” substitution: replace \(x^{2}\to xx_{1}\), \(y^{2}\to yy_{1}\), \(x\to\tfrac{x+x_{1}}{2}\), \(y\to\tfrac{y+y_{1}}{2}\). The normal is then the perpendicular through the same point. A line \(y=mx+c\) is tangent precisely when \(c\) satisfies a curve-specific condition: \(c=\dfrac{a}{m}\) for \(y^{2}=4ax\); \(c^{2}=a^{2}m^{2}+b^{2}\) for the ellipse; and \(c^{2}=a^{2}m^{2}-b^{2}\) for the hyperbola. The set of points from which the two tangents are mutually perpendicular is a tidy locus — the directrix for a parabola, and the director circle \(x^{2}+y^{2}=a^{2}+b^{2}\) (ellipse) or \(x^{2}+y^{2}=a^{2}-b^{2}\) (hyperbola).
8. Identifying a conic from the general equation
Given \(Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\) with \(B=0\) (no cross term, the level studied here), classify by the squared-term coefficients:
- \(A=C\) (and both present): a circle;
- exactly one of \(A,C\) is zero: a parabola;
- \(A\neq C\) with the same sign: an ellipse;
- \(A\) and \(C\) of opposite signs: a hyperbola.
Complete the square in \(x\) and in \(y\) to recover the centre and the standard form.
9. Degenerate conics
When the cutting plane passes through the vertex of the cone, the section collapses: an ellipse-type cut becomes a single point, a parabola-type cut becomes a line or a pair of parallel lines, and a hyperbola-type cut becomes a pair of intersecting lines. In equation terms, a right-hand side that should be positive turning into \(0\) (or negative) is the tell-tale sign of a degenerate or empty locus.
10. A few real-world appearances
Parabolic reflectors gather parallel rays to the focus (satellite dishes, head-lamps, solar cookers); planets and many comets travel in elliptical orbits with the Sun at a focus, and the “whispering gallery” and lithotripsy both exploit the ellipse's focus-to-focus reflection; cooling towers and long-range navigation systems use the hyperbola. Recognising the governing conic is usually the whole battle in these word problems.
Common exam traps
- Which axis is major? Compare the denominators first. The larger one sits under the major axis and tells you where the foci go.
- Right \(c\)-relation. Ellipse uses \(c^{2}=a^{2}-b^{2}\); hyperbola uses \(c^{2}=a^{2}+b^{2}\). Mixing them is the most common slip.
- Circle check before radius. A second-degree equation is a circle only if the \(x^{2}\) and \(y^{2}\) coefficients are equal (and there is no \(xy\) term). Normalise them to \(1\) before using \(\sqrt{g^{2}+f^{2}-c}\).
- Two values are allowed. Family-of-circles and tangent-from-a-point problems often have two valid answers — don't discard the second.
- Latus rectum direction. It is the focal chord perpendicular to the axis; its length is \(4a\) for a parabola and \(\dfrac{2b^{2}}{a}\) for an ellipse or hyperbola.
- Tangency conditions differ by a sign. Ellipse needs \(c^{2}=a^{2}m^{2}+b^{2}\), hyperbola needs \(c^{2}=a^{2}m^{2}-b^{2}\).