Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x + y = 3$. If $R$ and $r$ be the radius of circumcircle and incircle respectively of $\triangle ABC$, then $( R + r )$ is equal to:
(1) $\frac { 9 } { \sqrt { 2 } }$
(2) $7 \sqrt { 2 }$
(3) $2 \sqrt { 2 }$
(4) $3 \sqrt { 2 }$

In an isosceles triangle $ABC$, the vertex $A$ is $( 6,1 )$ and the equation of the base $BC$ is $2x + y = 4$. Let the point $B$ lie on the line $x + 3y = 7$. If $( \alpha , \beta )$ is the centroid of the triangle $ABC$, then $15( \alpha + \beta )$ is equal to

For $t \in (0, 2\pi)$, if $ABC$ is an equilateral triangle with vertices $A(\sin t, -\cos t)$, $B(\cos t, \sin t)$ and $C(a, b)$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $a^2 - b^2$ is equal to
(1) $\frac{8}{3}$
(2) $8$
(3) $\frac{77}{9}$
(4) $\frac{80}{9}$

Let the circumcentre of a triangle with vertices $A ( a , 3 ) , B ( b , 5 )$ and $C ( a , b ) , a b > 0$ be $P ( 1,1 )$. If the line $A P$ intersects the line $B C$ at the point $Q \left( k _ { 1 } , k _ { 2 } \right)$, then $k _ { 1 } + k _ { 2 }$ is equal to
(1) 2
(2) $\frac { 4 } { 7 }$
(3) $\frac { 2 } { 7 }$
(4) 4

The equations of the sides $AB , BC$ and $CA$ of a triangle $ABC$ are $2 x + y = 0 , x + p y = 15 a$ and $x - y = 3$ respectively. If its orthocentre is $( 2 , a ) , - \frac { 1 } { 2 } < a < 2$, then $p$ is equal to $\_\_\_\_$.

The equations of the sides $A B , B C$ and $C A$ of a triangle $A B C$ are $2 x + y = 0 , x + p y = 39$ and $x - y = 3$ respectively and $P ( 2,3 )$ is its circumcentre. Then which of the following is NOT true
(1) $( A C ) ^ { 2 } = 9 p$
(2) $( A C ) ^ { 2 } + p ^ { 2 } = 136$
(3) $32 <$ area $( \triangle A B C ) < 36$
(4) $34 <$ area $( \triangle A B C ) < 38$

The equations of the sides $AB , BC \& CA$ of a triangle $ABC$ are $2x + y = 0 , x + py = 21a ( a \neq 0 )$ and $x - y = 3$ respectively. Let $P ( 2 , a )$ be the centroid of the triangle $ABC$, then $( BC ) ^ { 2 }$ is equal to

The equations of sides $AB$ and $AC$ of a triangle $ABC$ are $( \lambda + 1 ) x + \lambda y = 4$ and $\lambda x + ( 1 - \lambda ) y + \lambda = 0$ respectively. Its vertex $A$ is on the $y$-axis and its orthocentre is $( 1,2 )$. The length of the tangent from the point $C$ to the part of the parabola $y ^ { 2 } = 6 x$ in the first quadrant is
(1) $\sqrt { 6 }$
(2) $2 \sqrt { 2 }$
(3) 2
(4) 4

Let $A ( 0,1 ) , B ( 1,1 )$ and $C ( 1,0 )$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y ^ { 2 } = 4 a x$ passing through $D$ is $( \alpha + \beta \sqrt { 2 } , 0 )$, where $\alpha$ and $\beta$ are rational numbers, then $\frac { \alpha } { \beta ^ { 2 } }$ is equal to
(1) 8
(2) 12
(3) 6
(4) $\frac { 9 } { 2 }$

Let $( \alpha , \beta )$ be the centroid of the triangle formed by the lines $15 x - y = 82$, $6 x - 5 y = - 4$ and $9 x + 4 y = 17$. Then $\alpha + 2 \beta$ and $2 \alpha - \beta$ are the roots of the equation
(1) $x ^ { 2 } - 7 x + 12 = 0$
(2) $x ^ { 2 } - 14 x + 48 = 0$
(3) $x ^ { 2 } - 13 x + 42 = 0$
(4) $x ^ { 2 } - 10 x + 25 = 0$

If $( \alpha , \beta )$ is the orthocenter of the triangle $ABC$ with vertices $A ( 3 , - 7 ) , B ( - 1,2 )$ and $C ( 4,5 )$, then $9 \alpha - 6 \beta + 60$ is equal to
(1) 25
(2) 35
(3) 30
(4) 40

If the orthocentre of the triangle, whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$ is $(\alpha, \beta)$, then the quadratic equation whose roots are $\alpha + 4\beta$ and $4\alpha + \beta$, is
(1) $x^2 - 19x + 90 = 0$
(2) $x^2 - 18x + 80 = 0$
(3) $x^2 - 22x + 120 = 0$
(4) $x^2 - 20x + 99 = 0$

Let $C ( \alpha , \beta )$ be the circumcentre of the triangle formed by the lines $4 x + 3 y = 69$, $4 y - 3 x = 17$, and $x + 7 y = 61$. Then $( \alpha - \beta ) ^ { 2 } + \alpha + \beta$ is equal to
(1) 18
(2) 17
(3) 15
(4) 16

If the orthocentre of the triangle formed by the lines $2 x + 3 y - 1 = 0 , x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $( 3,4 )$ and $( - 6 , - 8 )$, then the value of $| a - b |$ is $\_\_\_\_$

Two vertices of a triangle ABC are $\mathrm { A } ( 3 , - 1 )$ and $\mathrm { B } ( - 2,3 )$, and its orthocentre is $\mathrm { P } ( 1,1 )$. If the coordinates of the point C are $( \alpha , \beta )$ and the centre of the of the circle circumscribing the triangle PAB is $( \mathrm { h } , \mathrm { k } )$, then the value of $( \alpha + \beta ) + 2 ( \mathrm {~h} + \mathrm { k } )$ equals
(1) 5
(2) 81
(3) 15
(4) 51

Let $A ( 6,8 ) , B ( 10 \cos \alpha , - 10 \sin \alpha )$ and $C ( - 10 \sin \alpha , 10 \cos \alpha )$, be the vertices of a triangle. If $L ( a , 9 )$ and $G ( h , k )$ be its orthocenter and centroid respectively, then $( 5 a - 3 h + 6 k + 100 \sin 2 \alpha )$ is equal to $\_\_\_\_$

Let the area of a $\triangle P Q R$ with vertices $P ( 5,4 ) , Q ( - 2,4 )$ and $R ( a , b )$ be 35 square units. If its orthocenter and centroid are $O \left( 2 , \frac { 14 } { 5 } \right)$ and $C ( c , d )$ respectively, then $c + 2 d$ is equal to
(1) $\frac { 8 } { 3 }$
(2) $\frac { 7 } { 3 }$
(3) 2
(4) 3

Q83. Consider a triangle ABC having the vertices $\mathrm { A } ( 1,2 ) , \mathrm { B } ( \alpha , \beta )$ and $\mathrm { C } ( \gamma , \delta )$ and angles $\angle A B C = \frac { \pi } { 6 }$ and $\angle B A C = \frac { 2 \pi } { 3 }$. If the points B and C lie on the line $y = x + 4$, then $\alpha ^ { 2 } + \gamma ^ { 2 }$ is equal to $\_\_\_\_$

Q64. Let two straight lines drawn from the origin O intersect the line $3 x + 4 y = 12$ at the points P and Q such that $\triangle \mathrm { OPQ }$ is an isosceles triangle and $\angle \mathrm { POQ } = 90 ^ { \circ }$. If $l = \mathrm { OP } ^ { 2 } + \mathrm { PQ } ^ { 2 } + \mathrm { QO } ^ { 2 }$, then the greatest integer less than or equal to $l$ is :
(1) 42
(2) 46
(3) 44
(4) 48

Q84. If the orthocentre of the triangle formed by the lines $2 x + 3 y - 1 = 0 , x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $( 3,4 )$ and $( - 6 , - 8 )$, then the value of $| a - b |$ is $\_\_\_\_$

Q65. Two vertices of a triangle ABC are $\mathrm { A } ( 3 , - 1 )$ and $\mathrm { B } ( - 2,3 )$, and its orthocentre is $\mathrm { P } ( 1,1 )$. If the coordinates of the point C are $( \alpha , \beta )$ and the centre of the of the circle circumscribing the triangle PAB is $( \mathrm { h } , \mathrm { k } )$, then the value of $( \alpha + \beta ) + 2 ( \mathrm {~h} + \mathrm { k } )$ equals
(1) 5
(2) 81
(3) 15
(4) 51

Let side AB of an equilateral triangle ABC is given by $x + 2 \sqrt { 2 } y - 4 = 0$, where $A$ is on $x$-axis and $B$ is an $y$-axis. If origin $( 0,0 )$ is the orthocentre of the triangle ABC and vertex C is $( \alpha , \beta )$, then the value of $| \alpha - \sqrt { 2 \beta } |$ is (A) 0 (B) 2 (C) 4 (D) 6

(a) Show that the line $y = m x + c$ passes through the point $( 1,1 )$ if $c = 1 - m$.
(b) Let $L$ be a line with gradient $m > 0$, which passes through ( 1,1 ). Find the equation of the line $L ^ { \prime }$ which is perpendicular to $L$, and which passes through the point $( 1 , a )$, given $a \neq 1$.
(c) Find the area of the triangle which has ( 1,1 ) and ( $1 , a$ ) as two of its vertices and the intersection of $L$ and $L ^ { \prime }$ as the third vertex.
(d) For what value of $m$ is the triangle isosceles (two sides of equal length)?

A right-angled triangle has vertices at $( 2,3 ) , ( 9 , - 1 )$ and $( 5 , k )$.
Find the sum of all the possible values of $k$.

In the coordinate plane, let $\mathrm{O}(0,0)$ and $\mathrm{A}(0,1)$ be two points. Suppose two points $\mathrm{P}(p,0)$ and $\mathrm{Q}(q,0)$ on the $x$-axis satisfy both of the following conditions (i) and (ii).

• [(i)] $0 < p < 1$ and $p < q$
• [(ii)] Let $\mathrm{M}$ be the midpoint of segment $\mathrm{AP}$; then $\angle \mathrm{OAP} = \angle \mathrm{PMQ}$

(1) Express $q$ in terms of $p$.
(2) Find the value of $p$ such that $q = \dfrac{1}{3}$.
(3) Let $S$ be the area of $\triangle \mathrm{OAP}$ and $T$ be the area of $\triangle \mathrm{PMQ}$. Find the range of $p$ such that $S > T$.