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 $\_\_\_\_$ .

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) $\frac { \sqrt { 67 } } { 2 }$

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

Let $\left( 5 , \frac { a } { 4 } \right)$, be the circumcenter of a triangle with vertices $A ( a , - 2 ) , B ( a , 6 )$ and $C \left( \frac { a } { 4 } , - 2 \right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is
(1) 60
(2) 53
(3) 62
(4) 30

Let the parabola $y = x ^ { 2 } + \mathrm { p } x - 3$, meet the coordinate axes at the points $\mathrm { P } , \mathrm { Q }$ and R. If the circle C with centre at $( - 1 , - 1 )$ passes through the points $P , Q$ and $R$, then the area of $\triangle P Q R$ is:
(1) 7
(2) 4
(3) 6
(4) 5

Let $a$, $b$ and $c$ be positive real numbers. Consider a triangle ABC whose vertices are the three points $\mathrm{A}(a, 0)$, $\mathrm{B}(3, b)$ and $\mathrm{C}(0, c)$ on a plane with the coordinate system. Assume that the circumscribed circle of the triangle ABC passes through the origin $\mathrm{O}(0,0)$ and that $\angle\mathrm{BAC} = 60^\circ$.
(1) Since $\angle\mathrm{AOB} = \mathbf{AB}^\circ$, we obtain $b = \sqrt{\mathbf{C}}$.
(2) The equation of the circumscribed circle is
$$\left(x - \frac{a}{\mathbf{D}}\right)^2 + \left(y - \frac{c}{\mathbf{E}}\right)^2 = \frac{a^2 + c^2}{\mathbf{F}},$$
and $c$ can be expressed in terms of $a$ as $c = \sqrt{\mathbf{G}}\,(\mathbf{H} - a)$.
(3) Let D denote the point of intersection of the segment OB and the segment AC.
Set $\alpha = \angle\mathrm{OAC}$ and $\beta = \angle\mathrm{ADB}$. When $a = 2\sqrt{3}$, it follows that
$$\tan\alpha = \mathbf{I} - \sqrt{\mathbf{J}}, \quad \tan\beta = \mathbf{K}.$$

Q2 As shown in the figure, on an $xy$-plane whose origin is O, let us consider an isosceles triangle ABC satisfying $\mathrm { AB } = \mathrm { AC }$. Furthermore, suppose that side AB passes through $\mathrm { P } ( - 1,5 )$ and side AC passes through $\mathrm{Q}(3, 3)$.
Let us consider the radius of the inscribed circle of the triangle ABC.
Denote the straight line passing through the two points A and B by $\ell _ { 1 }$ and the straight line passing through the two points A and C by $\ell _ { 2 }$. When we denote the slope of $\ell _ { 1 }$ by $a$, the equations of $\ell _ { 1 }$ and $\ell _ { 2 }$ are
$$\begin{aligned} & \ell _ { 1 } : y = a x + a + \mathbf { M } , \\ & \ell _ { 2 } : y = - a x + \mathbf { N } a + \mathbf { O } . \end{aligned}$$
Denote the center and the radius of the inscribed circle by I and $r$, respectively. Then the coordinates of I are $\left( \mathbf { P } - \frac { \mathbf { Q } } { a } , r \right)$.
Hence $r$ can be expressed in terms of $a$ as
$$r = \frac { \mathbf { R } } { \mathbf { T } + \sqrt { \mathbf { S } } }$$
In particular, when $r = \frac { 5 } { 2 }$, the coordinates of vertex A are $\left( \frac { \mathbf { V } } { \mathbf{U} } , \frac { \mathbf { X Y } } { \mathbf { W } } \right)$.