Circles and conic sections for NDA

Conic sections look like four separate chapters, but they are one idea: a point moving so that its distance from a fixed point (the focus) is a fixed multiple e of its distance from a fixed line (the directrix). That multiple, the eccentricity, decides the curve. NDA questions here are mostly read-off questions: identify a, b and e from an equation, then write the focus, the directrix or the length of the latus rectum. A few test circles more deeply, through tangency or touching the axes.

Typical question types are:

• the centre and radius from a general equation;
• the equation of a circle from its diameter's ends, or with a given centre and a tangent;
• the condition for a line to touch a circle, or a circle to touch an axis;
• the focus, directrix, axis and latus rectum of a parabola;
• the eccentricity, foci and latus rectum of an ellipse or hyperbola;
• the equation of a conic from its foci and eccentricity;
• recognising which conic an equation represents.

The circle

• Standard form: (x − h)² + (y − k)² = r², centre (h, k), radius r.
• General form: x² + y² + 2gx + 2fy + c = 0, centre (−g, −f), radius √(g² + f² − c).

The general form is a real circle when g² + f² − c > 0 and a single point (a "point circle") when it equals 0. Notice what makes an equation a circle: the x² and y² coefficients are equal, and there is no xy term. If the coefficients are equal but not 1, divide through first.

Diameter form: if (x₁, y₁) and (x₂, y₂) are the ends of a diameter, the circle is (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0. It follows from the angle in a semicircle being 90°.

Special positions:

• passes through the origin when c = 0;
• touches the x-axis when g² = c, since then the radius equals |f|;
• touches the y-axis when f² = c.

Tangents to x² + y² = a². The line y = mx + c touches it when c² = a²(1 + m²). The reason is that the distance from the centre to the line, |c|/√(1 + m²), must equal the radius. The tangent at a point (x₁, y₁) on the circle is xx₁ + yy₁ = a².

For a point outside a circle S = 0, the length of the tangent from it is √S₁, where S₁ is the value of the left side of the general equation at that point.

The parabola (e = 1)

Here a > 0. The vertex is the origin, and the latus rectum (the chord through the focus perpendicular to the axis) has length 4a. The focal distance of a point (x, y) on y² = 4ax is x + a. Parametric point: (at², 2at).

The ellipse (0 < e < 1)

For x²/a² + y²/b² = 1 with a > b:

• b² = a²(1 − e²), so e = √(1 − b²/a²);
• foci (±ae, 0) and directrices x = ±a/e;
• major axis 2a along the x-axis, minor axis 2b;
• latus rectum 2b²/a;
• for any point on the ellipse, the sum of its distances from the two foci is 2a.

If b > a, the major axis lies along the y-axis: e = √(1 − a²/b²) and the foci are (0, ±be).

The hyperbola (e > 1)

For x²/a² − y²/b² = 1:

• b² = a²(e² − 1), so e = √(1 + b²/a²);
• foci (±ae, 0) and directrices x = ±a/e;
• transverse axis 2a, conjugate axis 2b;
• latus rectum 2b²/a;
• the difference of the distances from the foci is 2a;
• asymptotes y = ±(b/a)x.

When a = b the hyperbola is rectangular, with e = √2 and perpendicular asymptotes.

The four curves side by side

Recognising the curve (no xy term): equal x² and y² coefficients give a circle; unequal coefficients of the same sign give an ellipse; opposite signs give a hyperbola; only one squared term gives a parabola.

Worked NDA-style MCQs

Q1. The centre and radius of x² + y² − 6x + 4y − 12 = 0 are:
(a) (3, −2), 5 (b) (−3, 2), 5 (c) (3, −2), 25 (d) (6, −4), 5

2g = −6 and 2f = 4, so g = −3 and f = 2. The centre is (3, −2) and the radius is √(9 + 4 + 12) = 5. Answer: (a).

Q2. The circle on the join of (1, 2) and (5, 6) as diameter is:
(a) x² + y² − 6x − 8y + 17 = 0 (b) x² + y² + 6x + 8y + 17 = 0 (c) x² + y² − 6x − 8y + 7 = 0 (d) x² + y² − 3x − 4y + 17 = 0

(x − 1)(x − 5) + (y − 2)(y − 6) = x² − 6x + 5 + y² − 8y + 12. Check: centre (3, 4) and radius √(9 + 16 − 17) = 2√2, which is half the distance √32 between the two points. Answer: (a).

Q3. The line y = x + c touches x² + y² = 8 when c equals:
(a) ±2 (b) ±2√2 (c) ±4 (d) ±8

c² = a²(1 + m²) = 8 × 2 = 16, so c = ±4. Answer: (c).

Q4. A point on the parabola y² = 12x has x-coordinate 6. Its distance from the focus is:
(a) 6 (b) 9 (c) 12 (d) 3

4a = 12 gives a = 3, focus (3, 0) and directrix x = −3. The focal distance is x + a = 6 + 3 = 9. Answer: (b).

Q5. The length of the latus rectum of 9x² + 25y² = 225 is:
(a) 18/5 (b) 9/5 (c) 50/3 (d) 36/5

In standard form, x²/25 + y²/9 = 1, so a = 5 and b = 3. Latus rectum 2b²/a = 18/5. Along the way, e = √(1 − 9/25) = 4/5 and the foci are (±4, 0). Answer: (a).

Q6. The hyperbola with foci (±5, 0) and eccentricity 5/4 is:
(a) x²/9 − y²/16 = 1 (b) x²/16 − y²/9 = 1 (c) x²/16 + y²/9 = 1 (d) x²/25 − y²/16 = 1

ae = 5 and e = 5/4, so a = 4. b² = a²(e² − 1) = 16(25/16 − 1) = 9. Answer: (b).

Practice set

1. The centre and radius of x² + y² + 4x − 6y − 3 = 0 are: (a) (2, −3), 4 (b) (−2, 3), 4 (c) (−2, 3), 16 (d) (−2, 3), √10
2. The focus of y² = 8x is: (a) (8, 0) (b) (0, 2) (c) (2, 0) (d) (4, 0)
3. The eccentricity of x²/9 + y²/5 = 1 is: (a) 2/3 (b) √5/3 (c) 4/9 (d) 3/2
4. The foci of x²/16 + y²/25 = 1 are: (a) (±3, 0) (b) (0, ±3) (c) (0, ±4) (d) (±4, 0)
5. The directrix of x² = −16y is: (a) y = −4 (b) x = 4 (c) y = 4 (d) y = 16
6. The eccentricity of a rectangular hyperbola is: (a) 1 (b) 2 (c) √2 (d) 1/√2
7. x² + y² − 2x + 4y + 5 = 0 represents: (a) a circle of radius 1 (b) a point (c) no real locus (d) a circle of radius √5
8. The circle x² + y² − 4x − 6y + 4 = 0: (a) touches the x-axis (b) touches the y-axis (c) touches both axes (d) passes through the origin

Answers:

1. (b). g = 2 and f = −3; radius √(4 + 9 + 3) = 4.
2. (c). 4a = 8, so a = 2.
3. (a). √(1 − 5/9) = 2/3.
4. (b). Here b = 5 > a = 4, so the major axis is vertical; e = √(1 − 16/25) = 3/5 and be = 3.
5. (c). x² = −4ay with a = 4; the focus is (0, −4) and the directrix y = 4.
6. (c). a = b gives e = √(1 + 1).
7. (b). g² + f² − c = 1 + 4 − 5 = 0: a point circle at (1, −2).
8. (a). g² = 4 = c, so it touches the x-axis; f² = 9 ≠ 4, so it does not touch the y-axis. The radius is 3 = |f|.

What to do next

• Fill in the parabola table and the side-by-side table from memory.
• Solve 25 old NDA questions on circles and conics, always reducing the equation to standard form first.
• If slopes and distances are shaky, revise straight lines; then extend the same ideas to space in three-dimensional geometry.