Q65. Let $C$ be a circle with radius $\sqrt { 10 }$ units and centre at the origin. Let the line $x + y = 2$ intersects the circle C at the points P and Q . Let MN be a chord of C of length 2 unit and slope - 1 . Then, a distance (in units) between the chord PQ and the chord MN is
(1) $3 - \sqrt { 2 }$
(2) $\sqrt { 2 } + 1$
(3) $\sqrt { 2 } - 1$
(4) $2 - \sqrt { 3 }$

Q66. Let a circle $C$ of radius 1 and closer to the origin be such that the lines passing through the point $( 3,2 )$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $( 5,5 )$ is :
(1) $2 \sqrt { 2 }$
(2) $4 \sqrt { 2 }$
(3) 4
(4) 5

Q66. Let $A B C D$ and $A E F G$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment AB and the point F is on the diagonal AC . Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies:
(1) $r = 0$
(2) $2 r ^ { 2 } - 4 r + 1 = 0$
(3) $2 r ^ { 2 } - 8 r + 7 = 0$
(4) $r ^ { 2 } - 8 r + 8 = 0$

Q66. 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

Q66. If $\mathrm { P } ( 6,1 )$ be the orthocentre of the triangle whose vertices are $\mathrm { A } ( 5 , - 2 ) , \mathrm { B } ( 8,3 )$ and $\mathrm { C } ( \mathrm { h } , \mathrm { k } )$, then the point $C$ lies on the circle:
(1) $x ^ { 2 } + y ^ { 2 } - 61 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 52 = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 65 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 74 = 0$

Q66. Let the circles $C _ { 1 } : ( x - \alpha ) ^ { 2 } + ( y - \beta ) ^ { 2 } = r _ { 1 } ^ { 2 }$ and $C _ { 2 } : ( x - 8 ) ^ { 2 } + \left( y - \frac { 15 } { 2 } \right) ^ { 2 } = r _ { 2 } ^ { 2 }$ touch each other externally at the point $( 6,6 )$. If the point $( 6,6 )$ divides the line segment joining the centres of the circles $C _ { 1 }$ and $C _ { 2 }$ internally in the ratio $2 : 1$, then $( \alpha + \beta ) + 4 \left( r _ { 1 } ^ { 2 } + r _ { 2 } ^ { 2 } \right)$ equals
(1) 125
(2) 130
(3) 110
(4) 145

Q66. If the image of the point $( - 4,5 )$ in the line $x + 2 y = 2$ lies on the circle $( x + 4 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = r ^ { 2 }$, then r is equal to:
(1) 2
(2) 3
(3) 1
(4) 4

Q66. Let a circle passing through $( 2,0 )$ have its centre at the point $( h , k )$. Let $\left( x _ { c } , y _ { c } \right)$ be the point of intersection of the lines $3 x + 5 y = 1$ and $( 2 + c ) x + 5 c ^ { 2 } y = 1$. If $\mathrm { h } = \lim _ { \mathrm { c } \rightarrow 1 } x _ { \mathrm { c } }$ and $\mathrm { k } = \lim _ { \mathrm { c } \rightarrow 1 } y _ { \mathrm { c } }$, then the equation of the circle is :
(1) $25 x ^ { 2 } + 25 y ^ { 2 } - 2 x + 2 y - 60 = 0$
(2) $5 x ^ { 2 } + 5 y ^ { 2 } - 4 x + 2 y - 12 = 0$
(3) $5 x ^ { 2 } + 5 y ^ { 2 } - 4 x - 2 y - 12 = 0$
(4) $25 x ^ { 2 } + 25 y ^ { 2 } - 20 x + 2 y - 60 = 0$

Q67. A square is inscribed in the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 6 y + 30 = 0$. One side of this square is parallel to $y = x + 3$. If $\left( x _ { i } , y _ { i } \right)$ are the vertices of the square, then $\boldsymbol { \Sigma } \left( x _ { i } ^ { 2 } + y _ { i } ^ { 2 } \right)$ is equal to:
(1) 148
(2) 152
(3) 160
(4) 156

Q67. Let the line $2 x + 3 y - \mathrm { k } = 0 , \mathrm { k } > 0$, intersect the $x$-axis and $y$-axis at the points A and B , respectively. If the equation of the circle having the line segment AB as a diameter is $x ^ { 2 } + y ^ { 2 } - 3 x - 2 y = 0$ and the length of the latus rectum of the ellipse $x ^ { 2 } + 9 y ^ { 2 } = k ^ { 2 }$ is $\frac { m } { n }$, where $m$ and $n$ are coprime, then $2 \mathrm {~m} + \mathrm { n }$ is equal to
(1) 11
(2) 10
(3) 12
(4) 13

Q67. Let the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 2 ( x + y ) + 1 = 0$ and $C _ { 2 }$ be a circle having centre at $( - 1,0 )$ and radius 2 . If the line of the common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ intersects the $y$-axis at the point P , then the square of the distance of P from the centre of $\mathrm { C } _ { 1 }$ is :
(1) 2
(2) 1
(3) 4
(4) 6

Q67. Let $C$ be the circle of minimum area touching the parabola $y = 6 - x ^ { 2 }$ and the lines $y = \sqrt { 3 } | x |$. Then, which one of the following points lies on the circle $C$ ?
(1) $( 1,2 )$
(2) $( 1,1 )$
(3) $( 2,2 )$
(4) $( 2,4 )$

Q67. If the locus of the point, whose distances from the point $( 2,1 )$ and $( 1,3 )$ are in the ratio $5 : 4$, is $a x ^ { 2 } + b y ^ { 2 } + c x y + d x + e y + 170 = 0$, then the value of $a ^ { 2 } + 2 b + 3 c + 4 d + e$ is equal to $:$
(1) 37
(2) 437
(3) - 27
(4) 5

Q83. Let the centre of a circle, passing through the points $( 0,0 ) , ( 1,0 )$ and touching the circle $x ^ { 2 } + y ^ { 2 } = 9$, be $( h , k )$ - Then for all possible values of the coordinates of the centre $( h , k ) , 4 \left( h ^ { 2 } + k ^ { 2 } \right)$ is equal to

Q83. Let $A , B$ and $C$ be three points on the parabola $y ^ { 2 } = 6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left( \frac { A M \cdot B N } { C D } \right) ^ { 2 }$ is equal to $\_\_\_\_$

Q84. Suppose $A B$ is a focal chord of the parabola $y ^ { 2 } = 12 x$ of length $l$ and slope $\mathrm { m } < \sqrt { 3 }$. If the distance of the chord AB from the origin is d , then $l \mathrm {~d} ^ { 2 }$ is equal to $\_\_\_\_$

Q84. Let a line perpendicular to the line $2 x - y = 10$ touch the parabola $y ^ { 2 } = 4 ( x - 9 )$ at the point $P$. The distance of the point $P$ from the centre of the circle $x ^ { 2 } + y ^ { 2 } - 14 x - 8 y + 56 = 0$ is $\_\_\_\_$

Q84. Let S be the focus of the hyperbola $\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 5 } = 1$, on the positive $x$-axis. Let C be the circle with its centre at $A ( \sqrt { 6 } , \sqrt { 5 } )$ and passing through the point $S$. If $O$ is the origin and $S A B$ is a diameter of $C$, then the square of the area of the triangle OSB is equal to $\_\_\_\_$

Q84. Consider the circle $C : x ^ { 2 } + y ^ { 2 } = 4$ and the parabola $P : y ^ { 2 } = 8 x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $( \alpha , 0 )$ are bisected by the parabola $P$ is the interval $( p , q )$, then $( 2 q - p ) ^ { 2 }$ is equal to $\_\_\_\_$

Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16, 16)$. If $P(\alpha, \beta)$ divides this focal chord internally in the ratio $5:2$, then the minimum value of $\alpha + \beta$ is equal to:
(A) 7
(B) 5
(C) 22
(D) 16

Let $O$ be the vertex of the parabola $y^2 = 16x$. The locus of centroid of $\triangle OPA$ when $P$ lies on parabola and $A$ lies on $x$-axis and $\angle OPA = 90°$ is
(A) $y^2 = 8(3x - 16)$
(B) $9y^2 = 8(3x - 16)$
(C) $y^2 = 8(3x + 16)$
(D) $9y^2 = 8(3x + 16)$

The locus of point of intersection of tangent drawn to the circle $(x - 2)^{2} + (y - 3)^{2} = 16$, which substends an angle of $120^{\circ}$ is
(A) $3x^{2} + 3y^{2} + 12x + 18y - 25 = 0$ (B) $\mathrm{x}^{2} + \mathrm{y}^{2} - 12\mathrm{x} - 18\mathrm{y} - 25 = 0$ (C) $3x^{2} + 3y^{2} - 12x - 18y - 25 = 0$ (D) $x^{2} + y^{2} + 12x + 18y - 25 = 0$

the parabola $\mathbf { y } ^ { 2 } = 8 \mathbf { x }$ such that $\left( \frac { 7 } { 3 } , \frac { 4 } { 3 } \right)$ is the centrodd of the $( B C ) ^ { 2 }$ is equal to (A) 120 (B) 150 (C) 90

If $4 x ^ { 2 } + y ^ { 2 } \leq 52 , x , y \in I$ then number of ordered pairs ( $x , y$ ) is (A) 67 (B) 77 (C) 87 (D) 38

The line $y = x + 1$ intersects the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 1 } = 1$ at $A$ and $B$. Find the angle sub-stained by segment AB and centre of ellipse is\ (A) $\frac { \pi } { 2 } + \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$\ (B) $\frac { \pi } { 2 } + 2 \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$\ (C) $\frac { \pi } { 2 } + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$\ (D) $\frac { \pi } { 2 } - \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$