
How do you find the focus, vertex and directrix of ?
Answer
468.6k+ views
Hint: The equation is second degree in one variable and first degree in another variable . Such an equation is of a parabola. We can find the focus, vertex and directrix using the standard form of the equation of parabola . In this equation, the vertex is , focus is at a distance units from along the axis of the parabola, i.e. the focus is , and the directrix is a straight line perpendicular to the axis of the parabola situated at units from on the opposite side of the focus, i.e. equation of the directrix is given by, .
Complete step by step solution:
The given equation is . Since we have the equation in the second degree of exactly one variable, the given equation is of a parabola. To find the focus, vertex and the directrix we transform it into the standard form of the equation of the parabola .
We can write,
Therefore, , and .
The vertex of the given parabola is , i.e. the vertex is
The focus is , i.e. the focus is
The equation of the directrix is given by , i.e. the directrix is or .
Hence, for the given equation we get the vertex is , the focus is and the directrix is .
This is also shown in the graph,
Note: The vertex and focus of the parabola both lies on the axis of the parabola. The line of the directrix is perpendicular to the axis of the parabola. The distance of the vertex from the directrix is equal to the distance of the vertex from the focus, i.e. units. For an equation in second degree in we use the standard form of the equation as where the vertex is , the focus is and the directrix is .
Complete step by step solution:
The given equation is
We can write,
Therefore,
The vertex of the given parabola is
The focus is
The equation of the directrix is given by
Hence, for the given equation we get the vertex is
This is also shown in the graph,

Note: The vertex and focus of the parabola both lies on the axis of the parabola. The line of the directrix is perpendicular to the axis of the parabola. The distance of the vertex from the directrix is equal to the distance of the vertex from the focus, i.e.
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