Ch 5Two Dimensional Analytical Geometry-II — Class 12 Mathematics, concept-first.

Analytical (coordinate) geometry describes geometric objects — points, lines, circles, parabolas, ellipses, hyperbolas — using algebra on a Cartesian coordinate system.

The word circle is of Greek origin, and references to circles are found as far back as 1700 BCE. Circles are common in nature too — the Moon, the Sun, ripples in water — and the circle underlies the w…

(i) Centre at the origin. Let the centre be , the radius , and any point on the circle. Since for every position of , This is the equation of a circle with centre the origin and radius .

Diameter form (Theorem 5.2). Let and be the two ends of a diameter, and any point on the circle. Since an angle in a semicircle is a right angle, , so the chords and are perpendicular and the product…

Let the circle be (centre the origin, radius ) and the line .

Definition 5.2. A conic is the locus of a point that moves in a plane so that its distance from a fixed point (the focus) bears a constant ratio (the eccentricity, ) to its perpendicular distance from…

Let be the focus, the directrix, the eccentricity, and the moving point. By Definition 5.2, , i.e. , where and is the perpendicular distance from to the directrix.

Since for a parabola, a parabola is simply the set of points equidistant from a fixed focus and a fixed directrix.

An ellipse is the locus with : its distance from the focus is less than its distance from the directrix, scaled by .

A hyperbola is the locus with : its distance from the focus is greater than its distance from the directrix, scaled by .

Section 5.3 built every conic algebraically from the focus–directrix–eccentricity definition. There is also a purely geometric way to obtain the same four curves: slicing a double-napped cone (two ide…

Let the double-napped cone have its axis vertical, and consider a cutting plane that does not pass through the vertex, intersecting only one nappe:

If the cutting plane is instead tilted through the vertex of the double cone, the intersection collapses to a lower-dimensional degenerate conic:

Every conic — non-degenerate or degenerate — is some instance of . Rather than always completing the square, the following coefficient patterns identify the type directly (for the axis-aligned case ,…

Beyond the Cartesian equation and the focus–directrix description, a curve can also be traced by expressing both and as functions of a third variable : , . As varies, the point sweeps out the curve.

(i) Circle . Let be a point on the circle, making angle with the -axis, and the foot of the perpendicular from to the -axis. From right triangle : , .

A tangent to a curve is a straight line touching it at exactly one point; the normal at that point is the line through it perpendicular to the tangent.

(i) Tangent, cartesian form. For two points on : and , so , giving the chord's slope . Letting (so ) turns this chord into the tangent at , with slope :

The derivations mirror §5.6.1's chord-limit method for the parabola (left to the reader, as the textbook notes); only the results are needed for problem-solving, so they are collected here for direct…

(i) Parabola . Let be a tangent, touching at . Comparing it with the point-form tangent (same line, so coefficients proportional) gives , hence and (substituting into ) . So

The reflective and structural properties of the circle, parabola, ellipse and hyperbola — developed algebraically in §§5.2–5.6 — are exactly what makes each shape the natural engineering or physical c…

The parabola's headline application is as a reflector/receiver of light or radio waves: cross-sections of car headlights and flashlights are parabolas, formed as a paraboloid of revolution about the a…

By Kepler's laws, every planet in the solar system, including Earth, orbits the Sun on an ellipse with the Sun at one focus.

Some comets travel on hyperbolic (not elliptical) trajectories with the Sun at one focus — unlike elliptical-orbit comets, these pass the Sun only once and never return.

Reflective property. Light, sound, or radio waves originating at the focus of a parabola are reflected parallel to the axis (Fig.

Reflective property. The two lines from an ellipse's foci to any point on it make equal angles with the tangent at (Fig.

Reflective property. Exactly as for the ellipse, the two lines from a hyperbola's foci to a point on it make equal angles with the tangent at (Fig.

This 25-question multiple-choice set draws on every result in the chapter without introducing anything new: the family-of-circles idea and tangent/normal-to-a-circle results (§5.2), standard-form para…

Circle. Standard form (centre , radius ); general form (centre , radius ); family through a line–circle intersection ; diameter form ; tangent at : ; normal at : .

Sample papers and previous-year board questions for this subject.