Question 147
Um triângulo retângulo tem catetos medindo 6 cm e 8 cm. A área do círculo circunscrito a esse triângulo, em cm², é
(A) $16\pi$ (B) $25\pi$ (C) $36\pi$ (D) $49\pi$ (E) $100\pi$

21. (Multiple Choice) This problem includes four sub-problems A, B, C, and D. Please select two of them and answer in the corresponding areas. If more are done, the first two sub-problems will be graded. When solving, you should write out text explanations, proofs, or calculation steps.
A. [Elective 4-1: Geometric Proof Selection] (This problem is worth 10 points) In $\triangle A B C$, $A B = A C$, the chord $A E$ of the circumcircle O of $\triangle A B C$ intersects $B C$ at point D. Prove: $\triangle A B D \sim \triangle A E B$ [Figure]
B. [Elective 4-2: Matrices and Transformations] (This problem is worth 10 points) Given $x , y \in R$, the vector $\alpha = \left[ \begin{array} { c } 1 \\ - 1 \end{array} \right]$ is an eigenvector of the matrix $A = \left[ \begin{array} { c c } x & 1 \\ y & 0 \end{array} \right]$ corresponding to the eigenvalue $- 2$. Find the matrix A and its other eigenvalue.

C. [Elective 4-4: Coordinate Systems and Parametric Equations]

The polar equation of circle C is $\rho ^ { 2 } + 2 \sqrt { 2 } \rho \sin \left( \theta - \frac { \pi } { 4 } \right) - 4 = 0$. Find the radius of circle C.
D. [Elective 4-5: Inequalities Selection] Solve the inequality $x + | 2 x + 3 | \geq 3$

7. Given three points $A ( 1,0 ) , B ( 0 , \sqrt { 3 } ) , C ( 2 , \sqrt { 3 } )$, the distance from the circumcenter of $\triangle A B C$ [Figure]
to the origin is
A. $\frac { 5 } { 3 }$
B. $\frac { \sqrt { 21 } } { 3 }$
C. $\frac { 2 \sqrt { 5 } } { 3 }$
D. $\frac { 4 } { 3 }$

12. In $\triangle A B C$, the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given $10 \sin A - 5 \sin C = 2 \sqrt { 6 }$ and $\cos B = \frac { 1 } { 5 }$, then $\frac { c } { a } =$
$$\text { A. } \frac { 6 } { 7 } \quad \text{B.} \frac { 7 } { 6 } \quad \text{C.} \frac { 5 } { 6 } \quad \text{D.} \frac { 6 } { 5 }$$

Section II

II. Fill-in-the-Blank Questions: This section contains 4 questions, each worth 5 points, totaling 20 points. Write your answers on the answer sheet.

15. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ . Let $M$ be a point on $C$ in the first quadrant. If $\triangle M F _ { 1 } F _ { 2 }$ is an isosceles triangle, then the coordinates of $M$ are $\_\_\_\_$

15. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ , and let $M$ be a point on $C$ in the first quadrant. If $\triangle M F _ { 1 } F _ { 2 }$ is an isosceles triangle, then the coordinates of $M$ are $\_\_\_\_$ .

126. An isosceles trapezoid, under which condition can be inscribed in a circle?

1. [(1)] Two diameters perpendicular to each other
2. [(2)] One of the bases of the trapezoid equals one of the legs
3. [(3)] The line connecting the midpoints of the two legs passes through the intersection of the diameters
4. [(4)] The length of the segment connecting the midpoints of the two legs equals one of the legs

132- Which of the following quadrilaterals can be inscribed in a circle with diameter $(x+2)$?
[Figure: Trapezoid with sides labeled x, y, 4, 9 and angle $60^\circ$]

• [(1)] $\sqrt{51}$
• [(2)] $\sqrt{55}$
• [(3)] $\sqrt{57}$
• [(4)] $\sqrt{59}$

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 $x = AB$.

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.

A circle of radius $r$ is inscribed in a circular sector. The chord of the sector has length $a$. If the circle touches the chord and the two radii of the sector, find the relation between $a$ and $r$.
(A) $a = 8r/5$ (B) $a = 5r/8$ (C) $a = 4r/3$ (D) $a = 3r/4$

Let $P = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ and $Q = \left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)$ be two vertices of a regular polygon having 12 sides such that $PQ$ is a diameter of the circle circumscribing the polygon. Which of the following points is not a vertex of this polygon?
(A) $\left(\frac{\sqrt{3}-1}{2\sqrt{2}}, \frac{\sqrt{3}+1}{2\sqrt{2}}\right)$
(B) $\left(\frac{\sqrt{3}+1}{2\sqrt{2}}, \frac{\sqrt{3}-1}{2\sqrt{2}}\right)$
(C) $\left(\frac{\sqrt{3}+1}{2\sqrt{2}}, \frac{1-\sqrt{3}}{2\sqrt{2}}\right)$
(D) $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

In the following picture, $A B C$ is an isosceles triangle with an inscribed circle with center $O$. Let $P$ be the mid-point of $B C$. If $A B = A C = 15$ and $B C = 10$, then $O P$ equals:
(A) $\frac { \sqrt { 5 } } { \sqrt { 2 } }$
(B) $\frac { 5 } { \sqrt { 2 } }$
(C) $2 \sqrt { 5 }$
(D) $5 \sqrt { 2 }$.

Chords $A B$ and $C D$ of a circle intersect at right angle at the point $P$. If the lengths of $A P , P B , C P , P D$ are $2,6,3,4$ units respectively, then the radius of the circle is:
(A) 4
(B) $\frac { \sqrt { 65 } } { 2 }$
(C) $\frac { \sqrt { 66 } } { 2 }$
(D) $\frac { \sqrt { 67 } } { 2 }$

19. A pole stands vertically inside a triangular park $\triangle A B C$. If the angle of elevation of the top of the pole from each corner of the park is same, then in $\triangle \mathrm { ABC }$ the foot of the pole is at the :
(A) centroid
(B) circumcentre
(C) incentre
(D) orthocenter.

21. The incentre of the triangle with vertices $( 1 , \sqrt { 3 } ) , ( 0,0 )$ and $( 2,0 )$ is :
(A) $( 1 , \sqrt { } 3 / 2 )$
(B) $( 2 / 3,1 / \sqrt { } 3 )$
(C) $( 2 / 3 , \sqrt { } 3 / 2 )$
(D) $( 1,1 / \sqrt { } 3 )$

15. The centre of circle inscribed in square formed by the lines $x ^ { 2 } - 8 x + 12 = 0$ and $y ^ { 2 } - 14 y + 45 = 0$, is:
(a) $( 4,7 )$
(b) $( 7,4 )$
(c) $( 9,4 )$
(d) $( 4,9 )$

Tangent and normal are drawn at $P ( 16,16 )$ on the parabola $y ^ { 2 } = 16 x$, which intersect the axis of the parabola at $A \& B$, respectively. If $C$ is the center of the circle through the points $P , A \& B$ and $\angle C P B = \theta$, then a value of $\tan \theta$ is:
(1) $\frac { 4 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) 2
(4) 3

In a triangle $PQR$, the co-ordinates of the points $P$ and $Q$ are $(-2, 4)$ and $(4, -2)$ respectively. If the equation of the perpendicular bisector of $PR$ is $2x - y + 2 = 0$, then the centre of the circumcircle of the $\triangle PQR$ is:
(1) $(-1, 0)$
(2) $(-2, -2)$
(3) $(0, 2)$
(4) $(1, 4)$

Let the tangent to the circle $x ^ { 2 } + y ^ { 2 } = 25$ at the point $R ( 3,4 )$ meet $x$-axis and $y$-axis at point $P$ and $Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having centre at the incentre of the triangle $O P Q$, then $r ^ { 2 }$ is equal to
(1) $\frac { 529 } { 64 }$
(2) $\frac { 125 } { 72 }$
(3) $\frac { 625 } { 72 }$
(4) $\frac { 585 } { 66 }$

Consider a triangle having vertices $A ( - 2,3 ) , B ( 1,9 )$ and $C ( 3,8 )$. If a line $L$ passing through the circumcentre of triangle $ABC$, bisects line $BC$, and intersects $y$-axis at point $\left( 0 , \frac { \alpha } { 2 } \right)$, then the value of real number $\alpha$ is $\underline{\hspace{1cm}}$.

Let $A(\alpha, -2)$, $B(\alpha, 6)$ and $C\left(\frac{\alpha}{4}, -2\right)$ be vertices of a $\triangle ABC$. If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle ABC$, then which of the following is NOT correct about $\triangle ABC$
(1) area is 24
(2) perimeter is 25
(3) circumradius is 5
(4) inradius is 2

Let $O$ be the origin and $O P$ and $O Q$ be the tangents to the circle $x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 8 = 0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $O P Q$ passes through the point $\left( \alpha , \frac { 1 } { 2 } \right)$, then a value of $\alpha$ is
(1) $\frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) 1

A triangle is formed by the tangents at the point $( 2,2 )$ on the curves $y ^ { 2 } = 2 x$ and $x ^ { 2 } + y ^ { 2 } = 4 x$, and the line $\mathrm { x } + \mathrm { y } + 2 = 0$. If $r$ is the radius of its circumcircle, then $r ^ { 2 }$ is equal to $\_\_\_\_$

A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n ^ { 2 }$ is equal to
(1) 408
(2) 414
(3) 396
(4) 312