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## Chapter – 8

__Conic Section__

A parabola is the locus of a point which is equidistant from a fixed point (called the focus) and a fixed line (called the directrix).

### Standard Equation of a Parabola

The standard equations of a parabola are represented as ( y = ax² + bx + c ) for a parabola that opens upwards or downwards, and ( x = ay²+ by + c ) for a parabola that opens sideways. These equations are fundamental in describing the shape and characteristics of parabolic curves, such as the vertex, axis of symmetry, focus, and directrix.

## Conic Sections Class 12 Notes PDF

This PDF will provide the solutions of every question from the 2nd exercise of class 12 conic section which contains the exercise related to parabola. If you want the solutions of other exercises then you can select the exercise from the button given above.

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### Latus rectum, Focal distance and Focal chord

The figure given aside is a parabola whose vertex is A and focus is S. P is any point on the parabola.

- Focal Distance: The distance of any point on the parabola from the focus is called the focal distance or focal radius of the point. In the figure, SP is the focal distance and SP = PM = x+a
- Focal Chord: Any chord of the parabola passing through the focus is called a focal chord. In the figure, PSP’ is the focal chord of the parabola.
- Latus Rectum: The chord of the parabola passing through the focus and perpendicular to the axis is called the latus rectum of the parabola. In the figure, LSL’ is the latus rectum, L and L’ being the ends of the latus rectum which are the points of intersection of the parabola y² = 4ax and the line x = a. Solvingis entirely PLY the two equations, x = a y = plus/minus 2a

Therefore, the coordinates of the ends L and L’ of the latus rectum are (a, 2a) and (a, – 2a) respectively.

### Parabola with its Axis Parallel to the x-axis and Vertex at any Point

Let (h, k) be the vertex of a parabola whose axis is parallel to the x-axis, so that its focus is at (h + a, k) i.e. at a distance a’ from the focus. The directrix is parallel to y-axis and at a distance ‘a’ from the vertex, i.e. at a distance (h, – a) from the y-axis. So equation of the directrix is x – h + a = 0 If (x, y) be any point on the parabola, the distance of (x, y) from the focus (h + a, k) equals the distance of (x,y) from directrix x – h + a = 0.

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