Conics
General and Standard Form Equation of Conic Section
Submitted by vladimir.arzt on Sun, 03/17/2024 - 00:472010006309
Level:
C
Find the points of intersection of the circle \( x^2+y^2=16 \) and the line \( x-y+4=0 \).
\( [-4;0] \), \( [0;4] \)
\( [0;-4] \), \( [-4;0] \)
\( [4;0] \), \( [0;4] \)
\( [-4;0] \), \( [4;0] \)
\( [4;0] \), \( [0;-4] \)
2010006308
Level:
C
Find the equation of the circle, which passes through the points \( A=[-4;2] \), \( B=[1;3] \) and \( C=[2;-2] \).
\( (x+1)^2+y^2 = 13 \)
\( (x-1)^2+y^2 = 13 \)
\( x^2 + (y-1)^2 = 13 \)
\( x^2 + y^2 = 13x \)
\( x^2 + (y+1)^2 = 13 \)
2010006307
Level:
B
The equation of a parabola is given by \( 4x^2+16x-y+18=0 \). Find the equation of its directrix.
\( x=\frac{31}{16} \)
\( x=-\frac{33}{16} \)
\( x=\frac{33}{16} \)
\( x=-\frac{31}{16} \)
\( x=\frac{15}{8} \)
2010006306
Level:
B
The equation of a hyperbola with the center \( S=[1;-3] \), the focus \( F=[1;2] \), and the vertex \( A=[1;0] \) is given by:
\( \frac{(y+3)^2}{9}-\frac{(x-1)^2}{16} =1 \)
\( \frac{(x-1)^2}{16}-\frac{(y+3)^2}{9} =1 \)
\( \frac{(x-1)^2}{9}-\frac{(y+3)^2}{16} =1 \)
\( \frac{(y+3)^2}{16}-\frac{(x-1)^2}{9} =1 \)
\( \frac{(x+1)^2}{16}-\frac{(y-3)^2}{9} =1 \)
2010006305
Level:
B
Decide whether the equation \( 9x^2+4y^2-18x+8y+14=0 \) (in \( x \)-\( y \) plane) describes:
an empty set
an ellipse
a parabola
a hyperbola
a point
2010006304
Level:
B
The equation of a parabola is given by \( y^2-x-6y+10=0 \). What are the coordinates of its vertex?
\( [1;3] \)
\( [3;1] \)
\( [3;-1] \)
\( [-1;-3] \)
\( [-3;-1] \)
2010006303
Level:
B
A parabola passes through the points \( A=[5;0] \) and \( B=[-6;2] \) and it is symmetric with respect to \( x\)-axis. What are the coordinates of its vertex?
\( [5;0] \)
\( [6;-2] \)
\( [0;5] \)
\( [-6;2] \)
\( [-1;1] \)
2010006302
Level:
B
The equation of a hyperbola is given by \( -16x^2+9y^2-96x+108y+36=0\). The length of its semi-minor axis is:
\( 3 \)
\( 9\)
\( 4 \)
\( 16 \)
\( 5 \)