UFM Pure

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 Q11 5 marks Eccentricity or Asymptote Computation View

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

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 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 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. $|AE| \cdot |BE| \geq 18$

iran-konkur 2014 Q137 Equation Determination from Geometric Conditions View

137- The two lines $y = -2x$ and $y = 2x + 4$ are the asymptotes of a hyperbola, and $M\!\left(\dfrac{3}{2},\, 5\right)$ is one of its points. The distance between the two foci of this hyperbola is:
(1) $2\sqrt{3}$ (2) $2\sqrt{5}$ (3) $4\sqrt{3}$ (4) $4\sqrt{5}$
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iran-konkur 2015 Q137 Focal Distance and Point-on-Conic Metric Computation View

137- A parabola with focus $F(3,2)$ and a line with equation $x = -1$ intersect the $x$-axis at point $A$. What is the distance from point $A$ to the focus of the parabola?
(1) $2.75$ (2) $2.5$
(3) $2.75$ (4) $3$

iran-konkur 2015 Q138 Conic Identification and Conceptual Properties View

138- The rotation matrix $A$, with the relation $\begin{bmatrix} x \\ y \end{bmatrix} = A \cdot \begin{bmatrix} x' \\ y' \end{bmatrix}$, transforms the conic equation $5x^2 + 24xy - 2y^2 = 12$ into standard form with respect to $x'$ and $y'$. What is the tangent of the rotation angle?
(1) $\dfrac{2}{3}$ (2) $\dfrac{3}{4}$ (3) $\dfrac{4}{3}$ (4) $\dfrac{3}{2}$

iran-konkur 2016 Q137 Eccentricity or Asymptote Computation View

137- In the hyperbola $4x^2 - y^2 + 4y = 12$, $8x^2 - y^2 + 4y = 12$. What is the distance from a focus to an asymptote?
(1) $\sqrt{3}$ (2) $2$ (3) $2\sqrt{3}$ (4) $3$

iran-konkur 2018 Q136 Eccentricity or Asymptote Computation View

136- For which value of $a$, the asymptote $x = \dfrac{21}{8}$ of the conic $2y^2 - 12y + ax + 8 = 0$ holds?
(1) $12$ and $3$ (2) $16$ and $3$ (3) $12$ and $5$ (4) $16$ and $5$

iran-konkur 2018 Q137 Equation Determination from Geometric Conditions View

137- For which value of $a$, the distance between the foci of the hyperbola $3x^2 + 4y^2 + 16y + a = 0$ equals $2$?
(1) $2$ (2) $4$ (3) $6$ (4) $8$

iran-konkur 2019 Q134 Equation Determination from Geometric Conditions View

134- The parabola with focus $F(2,1)$ and directrix $x = 4$, the equation is which of the following?
(1) $y^2 - 2y + 4x = 11$ (2) $y^2 - 2y + 3x = 5$
(3) $x^2 - 4x + 4y = \circ$ (4) $x^2 - 6x + 2y = -4$

iran-konkur 2019 Q135 Focal Distance and Point-on-Conic Metric Computation View

135- In an ellipse with semi-major axis $2\sqrt{5}$ and 2 foci, the two foci and the two ends of the minor axis form a square. The sum of the squares of the focal radii of point $M$ on the ellipse is which of the following?
(1) $12$ (2) $16$ (3) $18$ (4) $20$

iran-konkur 2020 Q135 Focal Distance and Point-on-Conic Metric Computation View

135- In an ellipse with semi-axes $8$ and $2\sqrt{7}$, and foci $F$ and $F'$, a circle with diameter $F'F$ intersects the ellipse at point $M$. The distance from point $M$ to the nearest focus is:

[(1)] $4 - 3\sqrt{2}$
[(2)] $7.5$
[(3)] $4 - \sqrt{2}$
[(4)] $3$

iran-konkur 2020 Q136 Equation Determination from Geometric Conditions View

136- If the point $F(-0.25\ ,\ -2)$ is the focus of the parabola $y^2 + ay + bx + 1 = 0$, what is the smallest value of $b$?

[(1)] $-4$
[(2)] $-3$
[(3)] $-2$
[(4)] $2$

iran-konkur 2020 Q140 Conic Identification and Conceptual Properties View

140- The area of the graph of the curve $0 = 3x^2 + \sqrt{3}xy + 2y^2 - 10 = 0$ is:
\[ (1)\quad 6\pi \qquad (2)\quad 7\pi \qquad (3)\quad \frac{10\pi}{3} \qquad (4)\quad \frac{20\pi}{\sqrt{21}} \]

iran-konkur 2021 Q146 Equation Determination from Geometric Conditions View

146. The parabola $6 = 6y - 12y - (x-1)^2$ has vertex $F$ and focus $F'$. An ellipse has foci $F$ and $F'$ and eccentricity $0.6$. What is the distance from the center of the ellipse to the origin?
(1) $1$ (2) $\sqrt{2}$ (3) $\sqrt{3}$ (4) $2$

isi-entrance 2010 Q19 Locus and Trajectory Derivation View

Consider the branch of the rectangular hyperbola $xy = 1$ in the first quadrant. Let $P$ be a fixed point on this curve. The locus of the mid-point of the line segment joining $P$ and an arbitrary point $Q$ on the curve is part of
(a) A hyperbola
(b) A parabola
(c) An ellipse
(d) None of the above.

isi-entrance 2011 Q8 Identifying the Correct Graph of a Function View

The equation $x ^ { 3 } + y ^ { 3 } = x y ( 1 + x y )$ represents
(a) Two parabolas intersecting at two points
(b) Two parabolas touching at one point
(c) Two non-intersecting hyperbolas
(d) One parabola passing through the origin.

isi-entrance 2015 QB5 Intersection of Circles or Circle with Conic View

If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.

isi-entrance 2015 QB5 Intersection of Circles or Circle with Conic View

If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.

isi-entrance 2016 Q41 4 marks Tangent and Normal Line Problems View

Let $P$ be a point on the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C : P D$ equals
(A) 2
(B) $1 / 2$
(C) 4
(D) $1 / 4$

isi-entrance 2026 Q9 Tangent and Normal Line Problems View

Let $P$ be a point on the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C : P D$ equals
(a) 2 .
(B) $1 / 2$.
(C) 4 .
(D) $1 / 4$.

UFM Pure

Sequences and series, recurrence and convergence 563

Roots of polynomials 109

Polar coordinates 22

Conic sections 223

Taylor series 190

Hyperbolic functions 5

Integration using inverse trig and hyperbolic functions 17

Integration with Partial Fractions 28

Vectors: Lines & Planes 205

Vectors: Cross Product & Distances 40

First order differential equations (integrating factor) 217

Complex numbers 2 47

Second order differential equations 155

Systems of differential equations 8