sketch and answer for questions 34

Question

ScanSolve AI Tutor · Accepted Answer

Here are the construction steps for question 34:

34. Construct a parabola given the focus and directrix.

A parabola is defined as the locus of points that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Construction Steps:

Draw a line DD' to represent the directrix.
Mark a point F (the focus) not on the line DD'.
Draw a line through F perpendicular to DD'.
This line is the axis of the parabola. Let it intersect DD' at point K.
Locate the vertex V of the parabola.
The vertex V is the midpoint of the segment FK.
Choose several points $P_1, P_2, P_3, \dots$ on the axis, starting from V and moving away from F.
Through each point $P_i$ , draw a line $L_i$ perpendicular to the axis (and thus parallel to the directrix).
For each point $P_i$ , measure the distance $KP_i$ (the perpendicular distance from the directrix to $P_i$ ).
With F as the center and radius equal to $KP_i$ , draw arcs that intersect the line $L_i$ at two points, say $Q_i$ and $Q'_i$ .
These points $Q_i$ and $Q'_i$ are on the parabola because $FQ_i = KP_i$ .
Connect the vertex V and the points $Q_i, Q'_i$ with a smooth curve to form the parabola.
Sketch Description:
The sketch will show a horizontal line labeled DD' (the directrix). Above this line, there will be a point F (the focus). A vertical line, the axis of the parabola, will pass through F and be perpendicular to DD', intersecting DD' at point K. The vertex V will be marked as the midpoint of FK. Several horizontal lines ( $L_1, L_2, \dots$ ) parallel to DD' will be drawn above DD', intersecting the axis at points $P_1, P_2, \dots$ . For each $P_i$ , arcs will be drawn from F with radius $KP_i$ , intersecting the corresponding line $L_i$ at two points ( $Q_i, Q'_i$ ). A smooth, U-shaped curve, opening away from the directrix and passing through V and all the $Q_i, Q'_i$ points, will represent the parabola.

Sketch and answer for questions 34

Step-by-step solution

34. Construct a parabola given the focus and directrix.

Draw a line DD' to represent the directrix.

Draw a line through F perpendicular to DD'.

Locate the vertex V of the parabola.

Choose several points $P_1, P_2, P_3, \dots$ on the axis, starting from V and moving away from F.

Through each point $P_i$ , draw a line $L_i$ perpendicular to the axis (and thus parallel to the directrix).

For each point $P_i$ , measure the distance $KP_i$ (the perpendicular distance from the directrix to $P_i$ ).

With F as the center and radius equal to $KP_i$ , draw arcs that intersect the line $L_i$ at two points, say $Q_i$ and $Q'_i$ .

Connect the vertex V and the points $Q_i, Q'_i$ with a smooth curve to form the parabola.

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Sketch and answer for questions 34

Step-by-step solution

34. Construct a parabola given the focus and directrix.

Draw a line DD' to represent the directrix.

Draw a line through F perpendicular to DD'.

Locate the vertex V of the parabola.

Choose several points P1,P2,P3,…P_1, P_2, P_3, \dotsP1​,P2​,P3​,… on the axis, starting from V and moving away from F.

Through each point PiP_iPi​, draw a line LiL_iLi​ perpendicular to the axis (and thus parallel to the directrix).

For each point PiP_iPi​, measure the distance KPiKP_iKPi​ (the perpendicular distance from the directrix to PiP_iPi​).

With F as the center and radius equal to KPiKP_iKPi​, draw arcs that intersect the line LiL_iLi​ at two points, say QiQ_iQi​ and Qi′Q'_iQi′​.

Connect the vertex V and the points Qi,Qi′Q_i, Q'_iQi​,Qi′​ with a smooth curve to form the parabola.

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More Mathematics Questions

Choose several points $P_1, P_2, P_3, \dots$ on the axis, starting from V and moving away from F.

Through each point $P_i$ , draw a line $L_i$ perpendicular to the axis (and thus parallel to the directrix).

For each point $P_i$ , measure the distance $KP_i$ (the perpendicular distance from the directrix to $P_i$ ).

With F as the center and radius equal to $KP_i$ , draw arcs that intersect the line $L_i$ at two points, say $Q_i$ and $Q'_i$ .

Connect the vertex V and the points $Q_i, Q'_i$ with a smooth curve to form the parabola.