Exercises
You already know the four conics separately. Here is a quick reminder of the data you read from each standard equation.
You already know the four conics separately. The challenge now is to tell them apart quickly: given an equation or a statement, decide in seconds whether it is a circle, an ellipse, a hyperbola or a parabola, and solve the whole problem.
Which conic is it? Look at the general equation
Determine the coordinates of the centre and the radius of the circle with general equation $x^2 + y^2 - 8x + 10y - 8 = 0$.
Determine the coordinates of the centre and the radius of the circle with general equation $x^2 + y^2 - 8x + 10y - 8 = 0$.
Find the coordinates of the vertex, the focus, and the equation of the directrix of the parabola $y^2 = -20x$.
Find the coordinates of the vertex, the focus, and the equation of the directrix of the parabola $y^2 = -20x$.
Find the lengths of the semi-axes, the coordinates of the foci, and the eccentricity of the ellipse with equation $\frac{x^2}{100} + \frac{y^2}{64} = 1$.
Find the lengths of the semi-axes, the coordinates of the foci, and the eccentricity of the ellipse with equation $\frac{x^2}{100} + \frac{y^2}{64} = 1$.
Given the hyperbola with equation $\frac{x^2}{144} - \frac{y^2}{25} = 1$, determine its vertices, its foci, and the equations of its asymptotes.
Given the hyperbola with equation $\frac{x^2}{144} - \frac{y^2}{25} = 1$, determine its vertices, its foci, and the equations of its asymptotes.
Which conic is it? Look at the statement
When there is no equation, look at which condition the problem describes:
Write the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 12 = 0$ and $s: 5x + 12y - 10 = 0$.
Write the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 12 = 0$ and $s: 5x + 12y - 10 = 0$.
Find the equation of the perpendicular bisector of the line segment with endpoints $A(2, -3)$ and $B(-4, 5)$.
Find the equation of the perpendicular bisector of the line segment with endpoints $A(2, -3)$ and $B(-4, 5)$.
Identify the locus of points in the plane whose difference of distances, in absolute value, to the points $F_1(5, 0)$ and $F_2(-5, 0)$ is always equal to $8$. Write its…
Identify the locus of points in the plane whose difference of distances, in absolute value, to the points $F_1(5, 0)$ and $F_2(-5, 0)$ is always equal to $8$. Write its standard equation.
An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always double its distance to another fixed buoy at $B(4, 0)$. Find the…
An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always double its distance to another fixed buoy at $B(4, 0)$. Find the equation of the drone's path, simplify it, and identify the type of geometric curve it describes.
Multi-step context problems
Real-life context problems almost never give you $a$, $b$ or $c$ directly: you first have to translate the statement into those data.
A satellite describes an elliptical orbit around a planet, which is located at one of its foci. The minimum distance from the satellite to the planet (periapsis) is…
A satellite describes an elliptical orbit around a planet, which is located at one of its foci. The minimum distance from the satellite to the planet (periapsis) is $3 \text{ Mm}$ and the maximum distance (apoapsis) is $9 \text{ Mm}$. Determine the equation of the orbit and its eccentricity, assuming that the centre of the ellipse is at the origin and the focal axis lies on the x-axis.
The cross-section of a nuclear cooling tower has the shape of a hyperbola. The narrowest part of the tower is $40 \text{ m}$ wide and is at a height of $50 \text{ m}$…
The cross-section of a nuclear cooling tower has the shape of a hyperbola. The narrowest part of the tower is $40 \text{ m}$ wide and is at a height of $50 \text{ m}$ above the ground. If the base of the tower has a width of $80 \text{ m}$, find the equation of the hyperbola, assuming its centre coincides with the centre of the narrowest part.
The main cable of a suspension bridge forms a parabola. The two towers supporting it are $120 \text{ m}$ apart and have a height of $40 \text{ m}$ above the bridge deck.…
The main cable of a suspension bridge forms a parabola. The two towers supporting it are $120 \text{ m}$ apart and have a height of $40 \text{ m}$ above the bridge deck. The lowest point of the cable is $10 \text{ m}$ above the deck. Determine the equation of the parabola and calculate the height of the cable $30 \text{ m}$ from the centre of the bridge.
Conics and lines: intersections and tangents
To find the points where a line meets a conic, you substitute the line (solved for one variable) into the equation of the conic: you get a quadratic equation whose discriminant decides whether there are $2$, $1$ or $0$ solutions.
Tangent to a conic at a given point
Two circles: the radical axis
A laser beam strikes an elliptical mirror whose shape is given by the equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The point of incidence has x-coordinate $x = 3$ and…
A laser beam strikes an elliptical mirror whose shape is given by the equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The point of incidence has x-coordinate $x = 3$ and a positive y-coordinate. Find the equation of the tangent line to the mirror at that point.
Find the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Graph the situation, showing both loci and the points of…
Find the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Graph the situation, showing both loci and the points of intersection.
You can do it on paper and send a photo with your answer.
Find the locus of the points in the plane $P(x, y)$ such that the product of the gradients of the lines connecting them to the fixed points $A(-4, 0)$ and $B(4, 0)$ is…
Find the locus of the points in the plane $P(x, y)$ such that the product of the gradients of the lines connecting them to the fixed points $A(-4, 0)$ and $B(4, 0)$ is equal to $-\frac{9}{16}$. Identify the resulting conic, calculate its eccentricity, and sketch it.
Two telecommunications networks have coverage areas modelled by the circles $C_1: x^2 + y^2 - 6x + 8y = 0$ and $C_2: x^2 + y^2 - 2x - 4y + 1 = 0$. Find the equation of…
Two telecommunications networks have coverage areas modelled by the circles $C_1: x^2 + y^2 - 6x + 8y = 0$ and $C_2: x^2 + y^2 - 2x - 4y + 1 = 0$. Find the equation of the radical axis, prove analytically that the circles intersect at two points, and calculate the exact distance between their points of intersection.