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Solved problems for the chapter:

Introduction to Parabola

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Parabola

1. Characteristics of a parabola

2. Equation of the parabola

3. Equation of the parabola passing through the origin

3.1. The first form of the parabola

3.2. The second form of the parabola

3.3. The third form of the parabola

3.4. The fourth form of the parabola

4. Equation of parabola not passing through the origin

5. Forum

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Hyperbola

1. Hyperbola equation:

1.1. Equation of the hyperbola in the central position with the real axis on the abscissa axis

1.2. Equation of the hyperbola in the central position with the real axis on the ordinate axis:

1.3. Hyperbola equation in general form:

2. Hyperbola and line:

3. Forum

↓ Show theory [2] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:introduction_to_parabola/4686/

Example 1.

Given the curve :

Sketch the curve and obtain the properties of the equation.
The vertices of the curve are the end points of a circle. Obtain the equation of this circle.

Example 2.

Obtain the equation of the hyperbola centred at the origin with and .

Draw the graph of the hyperbola represented by .

The bands of the hyperbola measure 4 cm and 5 cm, the vertices are aligned vertically, and the intersection of the asymptotes of the hyperbola is the center of the circle . Obtain the equation of the hyperbola, the coordinates of all its characteristic points, and the equation of asymptotes.

What is the length of the chord that the parabola cuts from the line ?

What is the length of the chord that the parabola cuts off from the line ?

Determine whether the line is a tangent to the parabola .

10.

Do the line and the parabola have a contact point?

11.

Do the line and the parabola have any point of contact?

12.

Do the line and the parabola have a contact point?

13.

Do the line and the parabola have a contact point?

14.

The equation of the hyperbola is given. Obtain the vertices, coordinates of the center, foci and the equation of asymptotes of the hyperbola.

15.

Calculate the area of a triangle that has vertices at the foci of the hyperbola and the coordinate origin.

16.

Write down the equation of the hyperbola centered at the origin and passes through the points and .

17.

Obtain the characteristic (centre, themes, focal point, asymptote, poles) data of the curve represented by

18.

Obtain the equation of a hyperbola which has the same foci as the ellipse and the asymptotes .

19.

Determine the equation of a displaced hyperbola passing through the point , and the lines and are its asymptotes.

20.

Calculate the intersections of the hyperbola with the coordinate axes. Also determine the asymptote equations.

21.

Determine the tangent point between the line and the hyperbola .

22.

Obtain the equation of the hyperbola centred at the origin if one focus is at the point and the real pole measures 2 units. Determine the equations of the asymptotes and draw the hyperbola.

23.

Determine the equation of a hyperbola that has foci and , and its imaginary pole measures b = 5.

24.

Obtain the equation of the hyperbola centred at the origin, which has vertex and an asymptote .

25.

Determine the equation of a hyperbola that has foci in the vertices and vertices at the foci of the ellipse .

26.

Obtain the equation of the hyperbola that has vertices at the points of intersection of the circle with the axis x, and one focus is at the point . Write down its asymptotes as well. What angle do the asymptotes enclose with the abscissa?

27.

Move the hyperbola with the equation in parallel so that its centre is in the coordinate origin. Obtain the coordinates of the foci and the asymptote equation of the resulting hyperbola.

28.

A hyperbola and the ellipse have common foci. Determine the equation of the hyperbola.

29.

Obtain the equation of a hyperbola symmetric with respect to the coordinate axis and intersects the ordinate axis at y = - 3, and the asymptotes are the symmetries of the odd quadrants.

30.

Obtain the equation of the hyperbola that has foci on the abscissa, the asymptote and goes through the point .

31.

The hyperbola in the figure below has a focal point at and vertex at . Obtain the equation of the hyperbola and the equation of its asymptotes.

32.

In the coordinate system draw the hyperbola (also draw the asymptotes). Calculate and obtain the intersections of the hyperbola and the line .