UFM Pure

gaokao 2020 Q4 5 marks Focal Distance and Point-on-Conic Metric Computation View

Let $A$ be a point on the parabola $C : y ^ { 2 } = 2 p x$ ($p > 0$). The distance from point $A$ to the focus of $C$ is 12, and the distance to the $y$-axis is 9. Then $p =$
A. 2
B. 3
C. 6
D. 9

gaokao 2020 Q5 5 marks Equation Determination from Geometric Conditions View

Let $O$ be the origin of coordinates. The line $x = 2$ intersects the parabola $C : y ^ { 2 } = 2 p x ( p > 0 )$ at points $D$ and $E$ . If $O D \perp O E$ , then the focus coordinates of $C$ are
A. $\left( \frac { 1 } { 4 } , 0 \right)$
B. $\left( \frac { 1 } { 2 } , 0 \right)$
C. $( 1,0 )$
D. $( 2,0 )$

gaokao 2020 Q6 5 marks Locus and Trajectory Derivation View

In the plane, $A , B$ are two fixed points and $C$ is a moving point. If $\overrightarrow { A C } \cdot \overrightarrow { B C } = 1$, then the locus of point $C$ is
A. a circle
B. an ellipse
C. a parabola
D. a line

gaokao 2020 Q7 5 marks Equation Determination from Geometric Conditions View

Let $O$ be the origin of coordinates. The line $x = 2$ intersects the parabola $C : y ^ { 2 } = 2 p x ( p > 0 )$ at points $D , E$. If $O D \perp O E$, then the focus coordinates of $C$ are
A. $\left( \frac { 1 } { 4 } , 0 \right)$
B. $\left( \frac { 1 } { 2 } , 0 \right)$
C. $( 1,0 )$
D. $( 2,0 )$

gaokao 2020 Q8 5 marks Eccentricity or Asymptote Computation View

Let $O$ be the origin of coordinates. The line $x = a$ intersects the two asymptotes of the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ at points $D$ and $E$ respectively. If the area of $\triangle O D E$ is 8, then the minimum value of the focal distance of $C$ is
A． 4
B． 8
C． 16
D． 32

gaokao 2020 Q10 5 marks Chord Properties and Midpoint Problems View

For the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$, a line $l$ passes through the right focus $F$ and intersects the ellipse at points $P$ and $Q$, with $P$ in the second quadrant. Given $Q \left( x _ { Q } , y _ { Q } \right)$ and $Q ^ { \prime } \left( x _ { Q } ^ { \prime } , y _ { Q } ^ { \prime } \right)$ both on the ellipse, with $y _ { Q } + y _ { Q } ^ { \prime } = 0$ and $F Q ^ { \prime } \perp P Q$, find the equation of line $l$ as $\_\_\_\_$

gaokao 2020 Q11 5 marks Triangle or Quadrilateral Area and Perimeter with Foci View

Let $F _ { 1 } , F _ { 2 }$ be the two foci of the hyperbola $C : x ^ { 2 } - \frac { y ^ { 2 } } { 3 } = 1$ , $O$ be the origin, and point $P$ on $C$ with $| O P | = 2$ . The area of $\triangle P F _ { 1 } F _ { 2 }$ is
A. $\frac { 7 } { 2 }$
B. 3
C. $\frac { 5 } { 2 }$
D. 2

gaokao 2020 Q11 5 marks Eccentricity or Asymptote Computation View

For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with left and right foci $F _ { 1 } , F _ { 2 }$ respectively, the eccentricity is $\sqrt { 5 }$ . $P$ is a point on $C$ such that $F _ { 1 } P \perp F _ { 2 } P$ . If the area of $\triangle P F _ { 1 } F _ { 2 }$ is 4 , then $a =$
A. $1$
B. $2$
C. $4$
D. $8$

gaokao 2020 Q14 5 marks Eccentricity or Asymptote Computation View

For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$, one asymptote is $y = \sqrt { 2 } x$. Then the eccentricity of $C$ is $\_\_\_\_$ .

gaokao 2020 Q21 12 marks Fixed Point or Collinearity Proof for Line through Conic View

Let $A , B$ be the left and right vertices of the ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + y ^ { 2 } = 1 ( a > 1 )$ respectively, $G$ be the upper vertex of $E$ , and $\overrightarrow { A G } \cdot \overrightarrow { G B } = 8$ . $P$ is a moving point on the line $x = 6$ , the other intersection point of $P A$ with $E$ is $C$ , and the other intersection point of $P B$ with $E$ is $D$ .
(1) Find the equation of $E$ ;
(2) Prove that the line $C D$ passes through a fixed point.

gaokao 2020 Q21 12 marks Triangle or Quadrilateral Area and Perimeter with Foci View

Given the ellipse $C : \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { m ^ { 2 } } = 1 ( 0 < m < 5 )$ with eccentricity $\frac { \sqrt { 15 } } { 4 }$, where $A , B$ are the left and right vertices of $C$ respectively.
(1) Find the equation of $C$;
(2) If point $P$ is on $C$, point $Q$ is on the line $x = 6$, and $| B P | = | B Q | , B P \perp B Q$, find the area of $\triangle A P Q$ .

gaokao 2021 Q3 Eccentricity or Asymptote Computation View

3. For the parabola $y ^ { 2 } = 2 p x ( p > 0 )$, the distance from its focus to the line $y = x + 1$ is $\sqrt { 2 }$. Then $p =$
A. 1
B. 2
C. $2 \sqrt { 2 }$
D. 4
【Answer】B 【Solution】【Analysis】First determine the coordinates of the focus of the parabola, then use the point-to-line distance formula to find the value of $p$.
【Detailed Solution】The focus of the parabola has coordinates $\left( \frac { p } { 2 } , 0 \right)$. The distance from this point to the line $x - y + 1 = 0$ is: $\quad d = \frac { \left| \frac { p } { 2 } - 0 + 1 \right| } { \sqrt { 1 + 1 } } = \sqrt { 2 }$, Solving: $p = 2$ (we discard $p = -6$). Therefore, the answer is: B.

gaokao 2021 Q5 Eccentricity or Asymptote Computation View

5. Let $F_1, F_2$ be the two foci of hyperbola $C$. Let $P$ be a point on $C$, and $\angle F_1 P F_2 = 60°$, $|PF_1| = 3|PF_2|$. Then the eccentricity of $C$ is
A. $\frac{\sqrt{7}}{2}$
B. $\frac{\sqrt{13}}{2}$
C. $\sqrt{7}$
D. $\sqrt{13}$

gaokao 2022 Q3 5 marks Focal Distance and Point-on-Conic Metric Computation View

The parabola $y ^ { 2 } = 2 p x ( p > 0 )$ has its focus at a distance of $\sqrt { 2 }$ from the line $y = x + 1$. Then $p =$
A. 1
B. 2
C. $2 \sqrt { 2 }$
D. 4

gaokao 2022 Q5 5 marks Focal Distance and Point-on-Conic Metric Computation View

Let $F$ be the focus of the parabola $C: y^2 = 4x$, point $A$ is on $C$, point $B(3,0)$. If $|AF| = |BF|$, then $|AB| =$
A. $2$
B. $2\sqrt{2}$
C. $3$
D. $3\sqrt{2}$

gaokao 2022 Q6 5 marks Focal Distance and Point-on-Conic Metric Computation View

Let $F$ be the focus of the parabola $C : y ^ { 2 } = 4 x$ , point $A$ is on $C$ , point $B ( 3,0 )$ , if $| A F | = | B F |$ , then $| A B | =$
A. 2
B. $2 \sqrt { 2 }$
C. 3
D. $3 \sqrt { 2 }$

gaokao 2022 Q10 5 marks Eccentricity or Asymptote Computation View

For the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, let $A$ be the left vertex. Points $P$ and $Q$ are both on $C$ and symmetric about the $y$-axis. If the product of the slopes of $AP$ and $AQ$ is $\frac { 1 } { 4 }$, then the eccentricity of $C$ is:
A. $\frac { \sqrt { 3 } } { 2 }$
B. $\frac { \sqrt { 2 } } { 2 }$
C. $\frac { 1 } { 2 }$
D. $\frac { 1 } { 3 }$

gaokao 2022 Q11 5 marks Equation Determination from Geometric Conditions View

The ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { 1 } { 3 }$. Let $A _ { 1 } , A _ { 2 }$ be the left and right vertices of $C$ respectively, and $B$ be the upper vertex. If $\overrightarrow { B A _ { 1 } } \cdot \overrightarrow { B A _ { 2 } } = - 1$ , then the equation of $C$ is
A. $\frac { x ^ { 2 } } { 18 } + \frac { y ^ { 2 } } { 16 } = 1$
B. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$
C. $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 2 } = 1$
D. $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 32 } = 1$

gaokao 2022 Q11 5 marks Eccentricity or Asymptote Computation View

For an ellipse $C$, $F_1, F_2$ are its two foci, $M, N$ are two points on the ellipse. If $\cos \angle F_1NF_2 = \frac{3}{5}$, then the eccentricity of $C$ is
A. $\frac{1}{2}$
B. $\frac{3}{2}$
C. $\frac{\sqrt{13}}{2}$
D. $\frac{\sqrt{17}}{2}$

gaokao 2022 Q11 Tangent and Normal Line Problems View

11. Let $O$ be the origin. Point $A ( 1,1 )$ lies on the parabola $C : x ^ { 2 } = 2 p y$ ( $p > 0$ ). A line through point $B ( 0 , - 1 )$ intersects $C$ at points $P$ and $Q$. Then
A. The directrix of $C$ is $y = - 1$
B. Line $A B$ is tangent to $C$
C. $| O P | \cdot | O Q | > | O A | ^ { 2 }$
D. $| B P | \cdot | B Q | > | B A | ^ { 2 }$

gaokao 2022 Q14 5 marks Tangent and Normal Line Problems View

If the asymptotes of the hyperbola $y ^ { 2 } - \frac { x ^ { 2 } } { m ^ { 2 } } = 1 ( m > 0 )$ are tangent to the circle $x ^ { 2 } + y ^ { 2 } - 4 y + 3 = 0$, then $m =$ $\_\_\_\_$

gaokao 2022 Q15 5 marks Conic Identification and Conceptual Properties View

For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with eccentricity $e$, write out a value of $e$ that satisfies the condition ``the line $y = kx$ intersects the hyperbola at four distinct points'' and give one such value $\_\_\_\_$ .

gaokao 2022 Q19 12 marks Chord Properties and Midpoint Problems View

(1) Find the equation of $C$;
(2) Let the lines $MD$ and $ND$ intersect $C$ at another point $A$ and $B$ respectively. Denote the inclination angles of lines $MN$ and $AB$ as $\alpha$ and $\beta$ respectively. When $\alpha - \beta$ attains its maximum value, find the equation of line $AB$.

gaokao 2022 Q21 12 marks Equation Determination from Geometric Conditions View

An ellipse $E$ has its center at the origin, with axes of symmetry along the $x$-axis and $y$-axis, and passes through points $A ( 0 , - 2 ) , B \left( \frac { 3 } { 2 } , 1 \right)$.
(The remainder of this question was cut off in the source document.)

gaokao 2023 Q5 5 marks Chord Properties and Midpoint Problems View

Given the ellipse $\frac{x^2}{3}+y^2=1$ with left and right foci $F_1, F_2$ respectively, the line $y=x+m$ intersects $C$ at points $A$ and $B$. If the area of $\triangle F_1AB$ is 2 times the area of $\triangle F_2AB$, then $m=$
A. $\frac{2}{3}$
B. $\frac{\sqrt{2}}{3}$
C. $-\frac{\sqrt{2}}{3}$
D. $-\frac{2}{3}$

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