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Parabolas are one of the fundamental shapes in analytical geometry (also known as coordinate geometry) and belong to the family of conic sections. You encounter parabolas in many contexts: the path of a thrown object under gravity (ignoring air resistance), the shape of a satellite dish or flashlight reflector, and in solving quadratic equations by geometric means.
In mathematics, a parabola can be characterised in multiple equivalent ways:
- as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix);
- as a graph given by a quadratic function
(or its variants); - via a “vertex form”
(or rotated versions); - in the “geometric definition” relating distance formulas.
Parabola Problems and Solutions
Determine the equations of the following parabolas and indicate the values of their focal parameter, focus and directrix.
1. 
2. 
Determine the equations of the following parabolas and indicate the values of their focal parameter, focus and directrix.
1
2.
Determine the equations of the parabolas using the information given:
1. The directrix is x = −3 and the focus is (3, 0).
2. The directrix is y = 4 and the vertex is (0, 0).
3. The directrix is y = −5 and the focus is (0, 5).
4. The directrix is x = 2 and the focus is (−2, 0).
5. The focus is (2, 0) and the vertex is (0, 0).
6. The focus is (3, 2) and the vertex is (5, 2).
7. The focus is (−2, 5) and the vertex is (−2, 2).
8. The focus is (3, 4) and the vertex is (1, 4).
Determine the equations of the parabolas using the information given:
1 The directrix is x = −3 and the focus is (3, 0).
2. The directrix is y = 4 and the vertex is (0, 0).
3. The directrix is y = −5 and the focus is (0, 5).
4. The directrix is x = 2 and the focus is (−2, 0).
5. The focus is (2, 0) and the vertex is (0, 0).
6. The focus is (3, 2) and the vertex is (5, 2).
7. The focus is (−2, 5) and the vertex is (−2, 2).
8. The focus is (3, 4) and the vertex is (1, 4).
Calculate the vertex, focus and directrix of the following parabolas:
1 
2 
3 
Calculate the vertex, focus and directrix of the following parabolas:
2.
3.
Find the equation of the vertical parabola that passes through the points: A = (6, 1), B = (−2, 3) and C = (16, 6).
Find the equation of the vertical parabola that passes through the points: A = (6, 1), B = (−2, 3) and C = (16, 6).
Determine the equation of the parabola with a directrix of y = 0 and a focus at (2, 4).
Determine the equation of the parabola with a directrix of y = 0 and a focus at (2, 4).
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Determine the point(s) of intersection between the line r ≡ x + y − 5 = 0 and the parabola y² = 16x.
Determine the point(s) of intersection between the line r ≡ x + y − 5 = 0 and the parabola y² = 16x.
Find the equation of the horizontal parabola that passes through the point (3, 4) and has its vertex at (0, 0).
Find the equation of the horizontal parabola that passes through the point (3, 4) and has its vertex at (0, 0).
Determine the equation of the parabola with an axis parallel to the y-axis, vertex on the x-axis and which passes through the points A = (2, 3) and B = (−1, 12).
Determine the equation of the parabola with an axis parallel to the y-axis, vertex on the x-axis and which passes through the points A = (2, 3) and B = (−1, 12).
Axis parallel to the y-axis 
Vertex on the x-axis 
Determine the equation of the parabola with a directrix of x + y − 6 = 0 and a focus at (0, 0).
Determine the equation of the parabola with a directrix of x + y − 6 = 0 and a focus at (0, 0).
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There are errors with the answers. The 2y was squared during the substitutions, similar things occuring throughout the examples
In my opinion there are mistakes while substituting “y” into circle formula.
Am I going mad, or are all of the workings on the circle/line intersection examples wrong? Eg Example 4 there is a term of 5x missing in step 2.
It is clear and understandable .
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This is very nice but please show how to find the 4p
It’s already been in the formula, but if you reffering why the 4p was used in the formula try take chat with deepseek, it was a good gpt alternatives even better for math
Hi Enabel! Thanks for sharing this helpful advice!