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Chapter 15: Problem 1404

The latus rectum of a parabola is a line (a) through the focus (b) parallel to the directrix (c) perpendicular to the axis (d) all of these

Short Answer

Expert verified
The latus rectum of a parabola is a line segment that passes through the focus of the parabola, is parallel to the directrix, and is perpendicular to the axis of the parabola. Therefore, option (d) "all of these" is the correct answer, as it includes options (a), (b), and (c).

Step by step solution

01

Verify option (a)

Recall the definition of the latus rectum: it is a line segment that passes through the focus of the parabola. Therefore, option (a) is true.
02

Verify option (b)

The latus rectum is also parallel to the directrix. This can be visualized if we recall that both the latus rectum and the directrix are horizontal (or vertical) lines when considering the standard form of a parabola. Therefore, option (b) is true.
03

Verify option (c)

As the latus rectum passes through the focus and is parallel to the directrix, it is perpendicular to the axis of the parabola, which is a vertical line passing through the vertex and the focus. Therefore, option (c) is true.
04

Verify option (d)

Since options (a), (b), and (c) are all true, option (d) is also correct. The latus rectum of a parabola has all the properties mentioned in the exercise: it passes through the focus, is parallel to the directrix, and is perpendicular to the axis of the parabola.

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Most popular questions from this chapter

The point of intersection of the tangents at the ends of the latus rectum of the parabola \(\mathrm{y}^{2}=4 \mathrm{x}\) is ....... (a) \((-1,0)\) (b) \((1,0)\) (c) \((0,0)\) (d) \((0,1)\)

The axis of the parabola \(9 \mathrm{y}^{2}-16 \mathrm{x}-12 \mathrm{y}-57=0\) is (a) \(\mathrm{y}=0\) (b) \(16 \mathrm{x}+61=0\) (c) \(3 \mathrm{y}-2=0\) (d) \(3 \mathrm{y}-61=0\)

The angle between the tangents drawn from the point \((1,4)\) to the parabola \(y^{2}=4 x\) is (a) \((\pi / 2)\) (b) \((\pi / 3)\) (c) \((\pi / 4)\) (d) \((\pi / 6)\)

If the line \(\mathrm{x}-1=0\) is the directrix of the parabola \(\mathrm{y}^{2}-\mathrm{kx}+8=0\) then one of the values of \(\mathrm{k}\) is (a) 4 (b) \((1 / 8)\) (c) \((1 / 4)\) (d) 8

The equation of the chord of parabola \(\mathrm{y}^{2}=8 \mathrm{x}\). Which is bisected at the point \((2,-3)\) is (a) \(3 x+4 y-1=0\) (b) \(4 x+3 y+1=0\) (c) \(3 \mathrm{x}-4 \mathrm{y}+1=0\) (d) \(4 x-3 y-1=0\)
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