LFM Pure

gaokao 2021 Q16 Area and Geometric Measurement Involving Circles View

16. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$. Let $P , Q$ be two points on $C$ that are symmetric with respect to the origin, and $| P Q | = \left| F _ { 1 } F _ { 2 } \right|$. Then the area of quadrilateral $P F _ { 1 } Q F _ { 2 }$ is $\_\_\_\_$ .

III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17--21 are required questions that all students must answer. Questions 22 and 23 are optional questions; students should answer according to the requirements.

(A) Required Questions: Total 60 points.

gaokao 2021 Q20 12 marks Tangent Lines and Tangent Lengths View

20. (12 points) The parabola $C$ has its vertex at the origin $O$ and focus on the $x$-axis. The line $l: x =

gaokao 2021 Q21 Tangent Lines and Tangent Lengths View

21. The parabola $C$ has its vertex at the origin $O$ and its focus on the $x$-axis. The line $l : x = 1$ intersects $C$ at points $P , Q$, and $O P \perp O Q$. Given the point $M ( 2,0 )$, and circle $\odot M$ is tangent to $l$.
(1) Find the equations of $C$ and $\odot M$;
(2) Let $A _ { 1 } , A _ { 2 } , A _ { 3 }$ be three points on $C$. Lines $A _ { 1 } A _ { 2 }$ and $A _ { 1 } A _ { 3 }$ are both tangent to $\odot M$. Determine the positional relationship between line $A _ {

gaokao 2022 Q14 5 marks Circle Equation Derivation View

Point $M$ lies on the line $2 x + y - 1 = 0$. Both points $( 3,0 )$ and $( 0,1 )$ lie on circle $\odot M$. Then the equation of $\odot M$ is $\_\_\_\_$ .

gaokao 2022 Q14 5 marks Circle Equation Derivation View

The equation of a circle passing through three of the four points $(0,0), (4,0), (-1,1), (4,2)$ is $\_\_\_\_$.

gaokao 2022 Q14 Tangent Lines and Tangent Lengths View

14. Write the equation of a line that is tangent to both the circle $x ^ { 2 } + y ^ { 2 } = 1$ and the circle $( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 16$: $\_\_\_\_$ .

gaokao 2022 Q15 5 marks Circle Equation Derivation View

The equation of a circle passing through three of the four points $( 0,0 ) , ( 4,0 ) , ( - 1,1 ) , ( 4,2 )$ is $\_\_\_\_$ .

gaokao 2022 Q16 Chord Length and Chord Properties View

16. Given an ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ ( $a > b > 0$ ), with upper vertex $A$, two foci $F _ { 1 }$ and $F _ { 2 }$, and eccentricity $\frac { 1 } { 2 }$. A line through $F _ { 1 }$ perpendicular to $A F _ { 2 }$ intersects $C$ at points $D$ and $E$, with $| D E | = 6$. The perimeter of $\triangle A D E$ is $\_\_\_\_$ .
IV. Solution Questions: This section contains 6 questions, for a total of 70 points. Solutions should include explanations, proofs, or calculation steps.

gaokao 2023 Q8 5 marks Chord Length and Chord Properties View

The eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \ (a > 0 , b > 0)$ is $\sqrt{5}$ . One of its asymptotes intersects the circle $(x - 2)^{2} + (y - 3)^{2} = 1$ at points $A , B$ , then $|AB| =$
A. $\frac{1}{5}$
B. $\frac{\sqrt{5}}{5}$
C. $\frac{2\sqrt{5}}{5}$
D. $\frac{4\sqrt{5}}{5}$

gaokao 2023 Q12 Optimization on a Circle View

Circle $\odot O$ has radius 1. Line $PA$ is tangent to $\odot O$ at point $A$. Line $PB$ intersects $\odot O$ at points $B$ and $C$. $D$ is the midpoint of $BC$. If $| P O | = \sqrt { 2 }$, then the maximum value of $\overrightarrow { P A } \cdot \overrightarrow { P D }$ is
A. $\frac { 1 + \sqrt { 2 } } { 2 }$
B. $\frac { 1 + 2 \sqrt { 2 } } { 2 }$
C. $1 + \sqrt { 2 }$
D. $2 + \sqrt { 2 }$

gaokao 2024 Q3 4 marks Distance from Center to Line View

Find the distance from the center of the circle $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y = 0$ to the line $x - y + 2 = 0$

gaokao 2024 Q5 5 marks Circle-Related Locus Problems View

Given curve $C : x ^ { 2 } + y ^ { 2 } = 16 ( y > 0 )$, from any point $P$ on $C$, draw a perpendicular segment $P P ^ { \prime }$ to the $x$-axis, where $P ^ { \prime }$ is the foot of the perpendicular. The locus of the midpoint of segment $P P ^ { \prime }$ is
A. $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1 \quad ( y > 0 )$
B. $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 8 } = 1 \quad ( y > 0 )$
C. $\frac { y ^ { 2 } } { 16 } + \frac { x ^ { 2 } } { 4 } = 1 \quad ( y > 0 )$
D. $\frac { y ^ { 2 } } { 16 } + \frac { x ^ { 2 } } { 8 } = 1 \quad ( y > 0 )$

gaokao 2025 Q7 5 marks Circle-Line Intersection and Point Conditions View

If the circle $x^2 + (y+2)^2 = r^2$ $(r > 0)$ has exactly $2$ points at distance $1$ from the line $y = \sqrt{3}x + 2$, then the range of $r$ is
A. $(0,1)$
B. $(1,3)$
C. $(3, +\infty)$
D. $(0, +\infty)$

gaokao 2025 Q7 5 marks Circle-Line Intersection and Point Conditions View

If the circle $x^2 + (y+2)^2 = r^2$ $(r > 0)$ has exactly 2 points at distance 1 from the line $y = \sqrt{3}x + 2$, then the range of $r$ is
A. $(0,1)$
B. $(1,3)$
C. $(3, +\infty)$
D. $(0, +\infty)$

gaokao 2025 Q16 15 marks Area and Geometric Measurement Involving Circles View

Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{\sqrt{2}}{2}$ and major axis length 4.
(1) Find the equation of $C$;
(2) A line $l$ passing through the point $(0, -2)$ intersects $C$ at points $A, B$. Let $O$ be the origin. If the area of $\triangle OAB$ is $\sqrt{2}$, find $|AB|$.

grandes-ecoles 2013 QIII.A.1 Circle Equation Derivation View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y)$ in $\mathbb{R}^2$, we denote by $\varphi_A(x,y)$ the determinant of the matrix $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$ and we consider $\mathcal{CP}_A$ the curve in $\mathbb{R}^2$ defined by the equation: $\varphi_A(x,y) = 0$.
Verify that $\mathcal{CP}_A$ is a circle (we agree that a circle can be reduced to a point); we will call $\mathcal{CP}_A$ the eigenvalue circle of $A$. Specify its center $C_A$ and its radius $r_A$.

grandes-ecoles 2013 QIII.A.2 Circle-Line Intersection and Point Conditions View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, with $\varphi_A(x,y)$ the determinant of $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$ and $\mathcal{CP}_A$ the eigenvalue circle of $A$, specify, as a function of $A$, the cardinality of the intersection of $\mathcal{CP}_A$ with the $x$-axis $\mathbb{R} \times \{0\}$.

grandes-ecoles 2024 QVII Circle Equation Derivation View

Exercise VII

Consider the points $A$ and $B$ with respective coordinates in an orthonormal coordinate system: $$A ( 2 ; 0 ) \text { and } \mathrm { B } ( 0 ; - 4 ) \text {. }$$ VII-A- An equation of the line $( A B )$ is $2 x - y - 4 = 0$. VII-B- An equation of the perpendicular bisector of segment $[ A B ]$ is $x + 2 y + 3 = 0$. VII-C- An equation of the circle with diameter $[ A B ]$ is $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y = 0$. VII-D- The point with coordinates $( - 1 ; - 1 )$ belongs to the circle with diameter $[ A B ]$. VII-E- The line with equation $2 x - y + 1 = 0$ is tangent to the circle with diameter $[ A B ]$.
For each statement, indicate whether it is TRUE or FALSE.

isi-entrance None Q6 Circle-Related Locus Problems View

Tangents are drawn to a given circle from a point on a given straight line, which does not meet the given circle. Prove that the locus of the mid-point of the chord joining the two points of contact of the tangents with the circle is a circle.

isi-entrance 2009 Q7 Inscribed/Circumscribed Circle Computations View

Let $A, B, C, D, E$ be the vertices of a regular pentagon inscribed in a circle of radius $r$. Let $F$ be the midpoint of side $AB$. Find the circumradius $AO$ in terms of the side length $x = AB$.

isi-entrance 2010 Q5 Inscribed/Circumscribed Circle Computations View

A circle is inscribed in a triangle with sides $8, 15, 17$ cms. The radius of the circle in cms is
(a) 3
(b) $22/7$
(c) 4
(d) None of the above.

isi-entrance 2011 Q11 Circles Tangent to Each Other or to Axes View

Let $C _ { 1 } , C _ { 2 }$ and $C _ { 3 }$ be three circles lying in the same quadrant, each touching both the axes. Suppose also that $C _ { 1 }$ touches $C _ { 2 }$ and $C _ { 2 }$ touches $C _ { 3 }$. If the area of the smallest circle is 1 unit, then area of the largest circle is
(a) $\{ ( \sqrt{2} + 1 ) / ( \sqrt{2} - 1 ) \} ^ { 4 }$
(b) $( 1 + \sqrt{2} ) ^ { 2 }$
(c) $( 2 + \sqrt{2} ) ^ { 2 }$
(d) $2 ^ { 4 }$

isi-entrance 2012 Q23 Circle Equation Derivation View

Find the equation of the circle with $AB$ as diameter, where $A$ and $B$ are the intercepts of the line $2x + 3y = k$ on the coordinate axes.

isi-entrance 2013 Q21 4 marks Area and Geometric Measurement Involving Circles View

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units. Divide $S$ into $4n^2$ unit squares by drawing $2n - 1$ horizontal and $2n - 1$ vertical lines one unit apart. A circle of diameter $2n - 1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4n - 2$
(B) $4n$
(C) $8n - 4$
(D) $8n - 2$

isi-entrance 2013 Q23 4 marks Circles Tangent to Each Other or to Axes View

An isosceles triangle with base 6 cms. and base angles $30^\circ$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3\sqrt{3}/2$
(B) $\sqrt{3}/2$
(C) $\sqrt{3}$
(D) $4/\sqrt{3}$

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17--21 are required questions that all students must answer. Questions 22 and 23 are optional questions; students should answer according to the requirements.

Exercise VII

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144