Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x ^ { 2 } + y ^ { 2 } = 16$. If the centre of the locus of the point $P$, which divides the line segment $AB$ in the ratio $3 : 2$ is the point $C( \alpha , \beta )$, then the length of the line segment $AC$ is
(1) $\frac { 3 \sqrt { 5 } } { 5 }$
(2) $\frac { 4 \sqrt { 5 } } { 5 }$
(3) $\frac { 2 \sqrt { 5 } } { 5 }$
(4) $\frac { 6 \sqrt { 5 } } { 5 }$

A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio $2:3$, is a circle of radius
(1) $\frac{3}{5}\lambda$
(2) $\frac{2}{3}\lambda$
(3) $\frac{\sqrt{19}}{5}\lambda$
(4) $\frac{\sqrt{19}}{7}\lambda$

Let the locus of the mid points of the chords of circle $x^2 + (y-1)^2 = 1$ drawn from the origin intersect the line $x + y = 1$ at $P$ and $Q$. Then, the length of $PQ$ is:
(1) $\frac{1}{\sqrt{2}}$
(2) $\sqrt{2}$
(3) $\frac{1}{2}$
(4) 1

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

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 $\_\_\_\_$

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 $A , B$ 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 $\_\_\_\_$

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) $x^{2} + y^{2} + 12x + 18y - 25 = 0$

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) $9y^2 = 8(3x + 16)$

Let a circle passes through origin and the points $\mathrm { A } ( - \sqrt { 2 } \alpha , 0 ) , \mathrm { B } ( 0, \sqrt { 2 } \beta )$, where $\alpha$ and $\beta$ are non zero real parameters, such that its radius is 4 . Then the radius of locus of centroid of triangle $O A B$ is
(A) $\frac { 2 } { 3 }$
(B) $\frac { 4 } { 3 }$
(C) $\frac { 11 } { 3 }$

Let $C$ be the parabola $y = x ^ { 2 } + 1$ with origin O, and let P be the point $(a, 2a)$.
(1) Find the equation of the line passing through point P and tangent to parabola $C$.
The equation of the tangent line at point $(t, t ^ { 2 } + 1)$ on $C$ is
$$y = \square t x - t ^ { 2 } + 1$$
If this line passes through P, then $t$ satisfies the equation
$$t ^ { 2 } - \square a t + \text { エ } a - \text { オ } = 0$$
so $t = \square a -$ カ y. Therefore
When $a \neq$ ケ, there are 2 tangent lines to $C$ passing through P, and their equations are
$$y = ( \square a - \square ) x - \text { シ } a ^ { 2 } + \text { ス } a$$
a ⋯⋯⋯(1)
and
$$y = \text { せ } x$$
(2) Let $\ell$ be the line represented by equation (1) in (1). If the intersection of $\ell$ and the $y$-axis is $\mathrm { R } ( 0 , r )$, then $r = -$ シ $a ^ { 2 } +$ ス $a$. For $r > 0$,
ソ $< a <$ タ
タ and in this case, the area $S$ of triangle OPR is
$$S = \text { チ } \left( a ^ { \text {ツ } } - a \text { 园 } \right)$$
When ソ $< a <$ タ , examining the increase and decrease of $S$ , we find that $S$ attains its maximum value at $a = \frac { \text { ト } } { \text { ナ } }$
(3) When □ソ $< a <$ □タ, the area $T$ of the region enclosed by the parabola $C$, the line $\ell$ in (2), and the two lines $x = 0, x = a$ is
$$T = \frac { 1 } { \square } \text { ハ } a ^ { 3 } - \square a ^ { 2 } + \square$$
ト
In the range $\frac { 1 } { \text { ナ } } \leqq a <$ タ, $T$ is □ヘ.
In the range, $T$ is □ヘ. Choose the correct answer for □ヘ from the following options (0) through (5).
(0) decreasing (1) attains a local minimum but not a local maximum (2) increasing (3) attains a local maximum but not a local minimum (4) constant (5) attains both a local minimum and a local maximum