Plane Mathematics - Straight Line & Conic Section

Straight Lines

• Brief recall of two dimensional geometries from earlier classes

• Shifting of origin

• Slope of a line and angle between two lines

• Various forms of equations of a line −

  • Parallel to axis

  • Point-slope form

  • Slope-intercept form

  • Two-point form

  • Intercept form

  • Normal form

• General equation of a line

• Equation of family of lines passing through the point of intersection of two lines

• Distance of a point from a line

Conic Sections

• Sections of a cone −

  • Circles

  • Ellipse

  • Parabola

  • Hyperbola − a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section.

• Standard equations and simple properties of −

  • Parabola

  • Ellipse

  • Hyperbola

• Standard equation of a circle

SUMMARY

Straight Line

1. Slope (m) of a non-vertical line passing through the points (x1 , y1 ) and (x2 , y2 ).

2. If a line makes an angle á with the positive direction of x-axis, then the slope of the line is given by m = tan α, α ≠ 90°.

3. Slope of horizontal line is zero and slope of vertical line is undefined.

4. An acute angle (say θ) between lines L1 and L2 with slopes m1 and m2 is given by tanθ = | m2 - m1 / 1 + m1m2 | ,  1 + m1m2 ≠ 0.

5. Two lines are parallel if and only if their slopes are equal.

6. Two lines are perpendicular if and only if product of their slopes is –1.

7. Three points A, B and C are collinear, if and only if slope of AB = slope of BC.

8. Equation of the horizontal line having distance a from the x-axis is either y = a or y = – a.

9. Equation of the vertical line having distance b from the y-axis is either x = b or x = – b.

10. The point (x, y) lies on the line with slope m and through the fixed point (xo , yo ), if and only if its coordinates satisfy the equation y – yo = m (x – xo ).

11. The point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = mx + c .

12. If a line with slope m makes x-intercept d. Then equation of the line is y = m (x – d).

13. Equation of a line making intercepts a and b on the x-and y-axis, respectively, is x/a + y/b = 1.

14. The equation of the line having normal distance from origin p and angle between normal and the positive x-axis ω is given by  xcosω + ysinω = p .

15. Any equation of the form Ax + By + C = 0, with A and B are not zero, simultaneously, is called the general linear equation or general equation of a line.

Conic Section

In this Chapter the following concepts and generalisations are studied.

1. A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.

2. A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.

3. Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola.

4. An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.

5. Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse.

6. The eccentricity of an ellipse is the ratio between the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.

7. A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

8. Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.

9. The eccentricity of a hyperbola is the ratio of the distances from the centre of the hyperbola to one of the foci and to one of the vertices of the hyperbola.