The parabola as a locus · Grade 12 · 5 units
In The parabola as a locus (Grade 12 · 5 units) you practise 5 skills: The parameter, the focus and the directrix · The equation of a parabola · A line and a parabola · The tangent to a parabola · Loci that are parabolas. A parabola is the locus of all the points whose distance from a fixed point, the focus, equals their distance from a fixed line, the directrix. With the focus on the horizontal axis and the directrix at right angles to it, the same distance on either side of the origin, the result is the standard equation: y squared equals two p times x. The distance between the focus and the directrix is the absolute value of p. When p is positive, the focus is half of p to the right of the origin, the directrix is the same distance to the left, and the parabola opens to the right. When p is negative, everything flips and the parabola opens to the left. The parabola is symmetrical about the horizontal axis, and its vertex is at the origin.
To find the common points of a line and a parabola, substitute the line’s equation into the parabola’s and solve a quadratic equation. If the discriminant is positive there are two common points, if it is zero the line is a tangent to the parabola, and if it is negative there are no common points. To find the gradient of the tangent at a point on the parabola, differentiate both sides: the gradient is p divided by the vertical coordinate of the point. To find a locus, label a general point, write the condition with the distance formula and simplify. The same distance from a point and from a line that does not pass through it always gives a parabola: the point is the focus, the line is the directrix, and the vertex lies halfway between them. When the parabola moves, its focus, directrix and vertex move with it.
Tip: The number in front of x in the equation is twice p, not p itself. Halve it before you work out the focus.
What you learn
The parameter, the focus and the directrix
For example: The parabola y² = 2x is written in the form y² = 2px . What is p ?
Answer and solution
The answer: 1
Compare the coefficients: 2p = 2 , so p = 1 .
The equation of a parabola
For example: The focus of a parabola is F(−1, 0) and its directrix is the line x = 1 . What is the equation of the parabola?
- y² = −2x
- y² = 4x
- x² = −4y
- y² = −4x
Answer and solution
The answer: y² = −4x
The focus is (p / 2, 0) , so p / 2 = −1 and p = −2 . So 2p = −4 , and the equation is y² = −4x .
A line and a parabola
For example: The parabola y² = 12x and the line y = −2x − 6 are given. How many common points do they have?
- One common point
- Two common points
- No common points
Answer and solution
The answer: No common points
Substituting gives 4x² + 12x + 36 = 0 . Its discriminant is −432 , so there are no common points.
The tangent to a parabola
For example: What is the gradient of the tangent to the parabola y² = −2x at the point (−2, 2) ?
Answer and solution
The answer: −1/2
Here p = −1 , so the gradient is −1/2 .
Loci that are parabolas
For example: What is the locus of the points whose distance from the point A(6, −5) equals their distance from the line x = 0 ?
- A line
- A circle
- An ellipse
- A parabola
Answer and solution
The answer: A parabola
The same distance from a point and from a line that does not pass through it: the answer is a parabola.
Not on the site yet
Parts of this topic that are not built yet: The reflective property (enrichment) · The tangent to a parabola as a locus.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).