UFM Pure

gaokao 2023 Q10 5 marks Focal Chord and Parabola Segment Relations View

Let $O$ be the origin of coordinates. The line $y=-\sqrt{3}(x-1)$ passes through the focus of the parabola $C: y^2=2px$ $(p>0)$ and intersects $C$ at points $M$ and $N$. Let $l$ be the directrix of $C$. Then
A. $p=2$
B. $|MN|=\frac{8}{3}$
C. the circle with $MN$ as diameter is tangent to $l$
D. $\triangle OMN$ is an isosceles triangle

gaokao 2023 Q11 Chord Properties and Midpoint Problems View

Let $A$ and $B$ be two points on the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { 9 } = 1$. Which of the following four points could be the midpoint of segment $AB$?
A. $(1,1)$
B. $( - 1,2 )$
C. $( 1,3 )$
D. $( - 1 , - 4 )$

gaokao 2023 Q12 5 marks Focal Distance and Point-on-Conic Metric Computation View

The ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{6} = 1$ has foci $F_{1} , F_{2}$ and center $O$ . Let $P$ be a point on the ellipse. If $\cos \angle F_{1}PF_{2} = \frac{3}{5}$ , then $|OP| =$
A. $\frac{2}{5}$
B. $\frac{\sqrt{30}}{2}$
C. $\frac{3}{5}$
D. $\frac{\sqrt{35}}{2}$

gaokao 2023 Q13 Focal Distance and Point-on-Conic Metric Computation View

Given that point $A ( 1 , \sqrt { 5 } )$ lies on the parabola $C : y ^ { 2 } = 2 p x$, then the distance from $A$ to the directrix of $C$ is \_\_\_\_

gaokao 2023 Q20 12 marks Focal Chord and Parabola Segment Relations View

The line $x - 2y + 1 = 0$ intersects the parabola $y^{2} = 2px \ (p > 0)$ at points $A , B$ with $AB = 4\sqrt{15}$ .
(1) Find the value of $p$ ;
(2) Let $F$ be the focus of $y^{2} = 2px$ . Let $M , N$ be two points on the parabola such that $\overrightarrow{MF} \perp \overrightarrow{NF}$ . Find the minimum area of $\triangle MNF$ .

gaokao 2023 Q20 12 marks Equation Determination from Geometric Conditions View

Given the ellipse $C : \frac { y ^ { 2 } } { a ^ { 2 } } + \frac { x ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ with eccentricity $\frac { \sqrt { 5 } } { 3 }$, and point $A ( - 2,0 )$ lies on $C$.
(1) Find the equation of $C$.
(2) A line passing through point $( - 2,3 )$ intersects $C$ at points $P$ and $Q$. Lines $AP$ and $AQ$ intersect the $y$-axis at points $M$ and $N$ respectively. Prove that the midpoint of segment $MN$ is a fixed point.

gaokao 2024 Q10 6 marks Circle-Conic Interaction with Tangency or Intersection View

The directrix of parabola $C : y ^ { 2 } = 4 x$ is $l$. Let $P$ be a moving point on $C$. Draw a tangent line to circle $\odot A : x ^ { 2 } + ( y - 4 ) ^ { 2 } = 1$ through $P$, with $Q$ as the point of tangency. Draw a perpendicular from $P$ to line $l$, with $B$ as the foot of the perpendicular. Then
A. Line $l$ is tangent to $\odot A$
B. When $P , A , B$ are collinear, $| P Q | = \sqrt { 15 }$
C. When $| P B | = 2$, $P A \perp A B$
D. There are exactly 2 points $P$ satisfying $| P A | = | P B |$

gaokao 2024 Q11 5 marks Eccentricity or Asymptote Computation View

Given the parabola $y ^ { 2 } = 16 x$, the coordinates of the focus are \_\_\_\_.

gaokao 2024 Q11 6 marks Locus and Trajectory Derivation View

The shape ``$\varnothing$'' can be made into a beautiful ribbon. Consider it as part of the curve $C$ in the figure. It is known that $C$ passes through the origin $O$ , and points on $C$ satisfy: the abscissa is greater than $- 2$ , and the product of the distance to point $F ( 2,0 )$ and the distance to the line $x = a ( a < 0 )$ equals 4 . Then
A. $a = - 2$
B. The point $( 2 \sqrt { 2 } , 0 )$ is on $C$
C. The maximum ordinate of points on $C$ in the first quadrant is 1
D. When point $\left( x _ { 0 } , y _ { 0 } \right)$ is on $C$ , $y _ { 0 } \leqslant \frac { 4 } { x _ { 0 } + 2 }$

gaokao 2024 Q12 5 marks Eccentricity or Asymptote Computation View

Let the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ have left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 2 }$ parallel to the $y$-axis intersects $C$ at points $A$ and $B$ . If $\left| F _ { 1 } A \right| = 13 , | A B | = 10$ , then the eccentricity of $C$ is $\_\_\_\_$ .

gaokao 2024 Q13 5 marks Eccentricity or Asymptote Computation View

Given the hyperbola $\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$, find the slopes of lines passing through $( 3,0 )$ that have only one intersection point with the hyperbola \_\_\_\_.

gaokao 2024 Q16 15 marks Triangle or Quadrilateral Area and Perimeter with Foci View

(15 points) Given that $A ( 0,3 )$ and $P \left( 3 , \frac { 3 } { 2 } \right)$ are two points on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ .
(1) Find the eccentricity of $C$ ;
(2) If a line $l$ through $P$ intersects $C$ at another point $B$ , and the area of $\triangle A B P$ is 9 , find the equation of $l$ .

gaokao 2024 Q19 Chord Properties and Midpoint Problems View

Given the ellipse equation $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$. The foci and endpoints of the minor axis form a square with side length 2. A line $l$ passing through $( 0 , t ) ( t > \sqrt { 2 })$ intersects the ellipse at points $A, B$, and $C ( 0,1 )$. Connect $AC$ and it intersects the ellipse at $D$.
(1) Find the equation of the ellipse and its eccentricity;
(2) If the slope of line $BD$ is 0, find $t$.

gaokao 2024 Q19 Locus and Trajectory Derivation View

Given hyperbola $C : x ^ { 2 } - y ^ { 2 } = m ( m > 0 )$, point $P _ { 1 } ( 5,4 )$ is on $C$, and $k$ is a constant with $0 < k < 1$. Through $P_n$ on the right branch of $C$, draw a line with slope $k$; this line intersects $C$ at another point $Q_n$ on the left branch. The reflection of $Q_n$ across the $y$-axis gives $P_{n+1}$ on the right branch.
(1) Find the coordinates of $P_2$ when $k = \frac{1}{2}$.
(2) Prove that $\{x_n - y_n\}$ is a geometric sequence.
(3) Prove that $S_n = \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1})$ is a constant independent of $n$.

gaokao 2025 Q3 5 marks Eccentricity or Asymptote Computation View

If the imaginary axis length of hyperbola $C$ is $\sqrt{7}$ times the real axis length, then the eccentricity of $C$ is
A. $\sqrt{2}$
B. $2$
C. $\sqrt{7}$
D. $2\sqrt{2}$

gaokao 2025 Q3 5 marks Eccentricity or Asymptote Computation View

If the imaginary axis length of hyperbola $C$ is $\sqrt{7}$ times the real axis length, then the eccentricity of $C$ is
A. $\sqrt{2}$
B. $2$
C. $\sqrt{7}$
D. $2\sqrt{2}$

gaokao 2025 Q6 5 marks Focal Distance and Point-on-Conic Metric Computation View

Let the parabola $C: y^2 = 2px$ $(p > 0)$ have focus $F$. Point $A$ is on $C$. A perpendicular is drawn from $A$ to the directrix of $C$, with foot $B$. If the equation of line $BF$ is $y = -2x + 2$, then $|AF| = $ ( )
A. $3$
B. $4$
C. $5$
D. $6$

gaokao 2025 Q10 6 marks Focal Chord and Parabola Segment Relations View

Let the focus of parabola $C: y^2 = 6x$ be $F$. A line through $F$ intersects $C$ at $A$ and $B$. A line through $F$ perpendicular to $AB$ intersects the directrix $l: x = -\frac{3}{2}$ at $E$. From point $A$, draw a perpendicular to the directrix $l$ with foot $D$. Then
A. $|AD| = |AF|$
B. $|AE| = |AB|$
C. $|AB| \geq 6$
D. $|AE| \cdot |BE| \geq 18$

gaokao 2025 Q10 6 marks Focal Chord and Parabola Segment Relations View

Let the focus of parabola $C: y^2 = 6x$ be $F$. A line through $F$ intersects $C$ at $A$ and $B$. A perpendicular from $A$ to the line $l: x = -\frac{3}{2}$ meets it at $D$. A line through $F$ perpendicular to $AB$ meets $l$ at $E$. Then
A. $|AD| = |AF|$
B. $|AE| = |AB|$
C. $|AB| \geq 6$
D. $|AE| \cdot |BE| \geq 18$

gaokao 2025 Q11 6 marks Eccentricity or Asymptote Computation View

For the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, let $F_1, F_2$ be the left and right foci respectively, and $A_1, A_2$ be the left and right vertices respectively. The circle with diameter $F_1F_2$ intersects one asymptote of $C$ at points $M, N$, and $\angle NA_1M = \frac{5\pi}{6}$, then
A. $\angle A_1MA_2 = \frac{\pi}{6}$
B. $|MA_1| = 2|MA_2|$
C. The eccentricity of $C$ is $\sqrt{13}$
D. When $a = \sqrt{2}$, the area of quadrilateral $NA_1MA_2$ is $8\sqrt{3}$

gaokao 2025 Q18 17 marks Optimization on Conics View

Let the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ have eccentricity $\frac{2\sqrt{2}}{3}$, with lower vertex $A$ and right vertex $B$, $|AB| = \sqrt{10}$.
(1) Find the standard equation of the ellipse.
(2) Let $P$ be a moving point not on the $y$-axis, and let $R$ be a point on the ray $AP$ satisfying $|AR| \cdot |AP| = 3$.
(i) If $P(m, n)$, find the coordinates of point $R$ (expressed in terms of $m, n$).
(ii) Let $O$ be the origin, and $Q$ be a moving point on $C$. The slope of line $OR$ is 3 times the slope of line $OP$. Find the maximum value of $|PQ|$.

gaokao 2025 Q18 17 marks Optimization on Conics View

(17 points) Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{2\sqrt{2}}{3}$, lower vertex $A$, right vertex $B$, and $|AB| = \sqrt{10}$.
(1) Find the equation of $C$.
(2) Let $P$ be a moving point not on the $y$-axis, and let $R$ be a point on the ray $AP$ satisfying $|AP| \cdot |AR| = 3$.
(i) If $P(m, n)$, find the coordinates of $R$ (expressed in terms of $m, n$).
(ii) Let $O$ be the origin, and $Q$ be a moving point on $C$. The slope of line $OR$ is 3 times the slope of line $OP$. Find the maximum value of $|PQ|$.

grandes-ecoles 2012 QII.B Conic Identification and Conceptual Properties View

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We denote by $\lambda _ { 1 } \leqslant \lambda _ { 2 }$ the eigenvalues of $A$.
We consider the set $\Gamma \subset \mathbb { R } ^ { 2 }$ defined by the equation $\langle A X , X \rangle = 1$.
II.B.1) Characterize the conditions on the $\lambda _ { i }$ for which this set is: a) empty; b) the union of two lines; c) an ellipse; d) a hyperbola.
II.B.2) Represent on the same figure the sets $\Gamma$ obtained for $A$ diagonal with $\lambda _ { 1 } \in \{ - 4 , - 1,0,1 / 4,1 \}$ and $\lambda _ { 2 } = 1$.

grandes-ecoles 2013 QV.B.2 Locus and Trajectory Derivation View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
Specify the intersection $Z_A$ of $\mathcal{H}_A$ with the plane with equation $x = (a+d)/2$.

grandes-ecoles 2013 QV.C.1 Locus and Trajectory Derivation View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, and let $Z_A$ be the intersection of $\mathcal{H}_A$ with the plane $x = (a+d)/2$.
If the matrix $A$ has two non-real eigenvalues, how can one see the eigenvalues of $A$ on $\mathcal{H}_A$? (One may consider the intersection of $Z_A$ with the plane with equation $y = 0$.) Can one see a basis of eigenvectors using $\mathcal{H}_A$?

UFM Pure

Sequences and series, recurrence and convergence 375

Roots of polynomials 146

Polar coordinates 22

Conic sections 291

Taylor series 183

Hyperbolic functions 10

Integration using inverse trig and hyperbolic functions 6

Integration with Partial Fractions 7

Vectors: Lines & Planes 293

Vectors: Cross Product & Distances 20

First order differential equations (integrating factor) 60

Complex numbers 2 113

Second order differential equations 127

Systems of differential equations 41