35. The maximum value of $( \cos a 1 ) \cdot ( \cos a 2 ) \cdot \ldots \cdot \cdot ( \cos a n )$, under the restrictions $0 \leq \mathrm { a } 1 , \mathrm { a } 2 , \ldots \ldots , \mathrm { an } \leq \pi / 2$ and $( \cot \mathrm { a } 1 )$. ( $\cot \mathrm { a } 2$ ). ..... ( $\cot \mathrm { an } ) = 1$ is:
(A) $1 / 2 n / 2$
(B) $1 / 2 n$
(C) $1 / 2 n$
(D) 1

6. Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joininɛ centre of the ellipse of the point of contact meet on the corresponding directrix.

15. If the tangent at the point $P$ on the circle $x ^ { 2 } + y ^ { 2 } + 6 x + 6 y = 2$ meets the straight line $5 x + 2 y = 6$ at a point $Q$ on the $y$-axis, then the length of $P Q$ is
(A) 4
(B) $2 \sqrt { } 5$
(C) 5
(D) $3 \sqrt { 5 }$

16. If $a > 2 b > 0$ then the positive value of $m$ for which $y = m x - b \sqrt { } \left( 1 + m ^ { 2 } \right)$ is $a$ common tangent to $x ^ { 2 } + y ^ { 2 } = b ^ { 2 }$ and $( x - a ) ^ { 2 } + y ^ { 2 } = b ^ { 2 }$ is
(A) $2 b / \sqrt { } \left( a ^ { 2 } - 4 b ^ { 2 } \right)$
(B) $\quad \sqrt { } \left( a ^ { 2 } - 4 b ^ { 2 } \right) / 2 b$

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(C) $2 \mathrm {~b} / ( \mathrm { a } - 2 \mathrm {~b} )$
(D) $\mathrm { b } / ( \mathrm { a } - 2 \mathrm {~b} )$

For the circle $x ^ { 2 } + y ^ { 2 } = t ^ { 2 }$, find the value of $r$ for which the area enclosed by the tangents drawn from the point $\mathrm { P } ( 6,8 )$ to the circle and the chord of contact is maximum.

Normals are drawn from the point $P$ with slopes $m _ { 1 } , m _ { 2 } , m _ { 3 }$ to the parabola $y ^ { 2 } = 4 x$. If locus of $P$ with $\mathrm { m } _ { 1 } \mathrm { m} _ { 2 } = \mathrm { a }$ is a part of the parabola itself then finda.

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) $( 4,9 )$

16. The focal chord to $y ^ { 2 } = 16 x$ is tangent to $( x - 6 ) ^ { 2 } + y ^ { 2 } = 2$, then the possible values of the slope of this chord, are :
(a) $\{ - 1,1 \}$
(b) $\{ - 2,2 \}$
(c) $\{ - 2,1 / 2 \}$
(d) $\{ 2,1 / 2 \}$

12. A circle touches the line $2 x + 3 y + 1 = 0$ at the point $( 1 , - 1 )$ and is orthogonal to the circle which has the line segment having end points $( 0 , - 1 )$ and $( - 2,3 )$ as the diameter.
Sol. Let the circle with tangent $2 \mathrm { x } + 3 \mathrm { y } + 1 = 0$ at $( 1 , - 1 )$ be $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + \lambda ( 2 x + 3 y + 1 ) = 0$ or $x ^ { 2 } + y ^ { 2 } + x ( 2 \lambda - 2 ) + y ( 3 \lambda + 2 ) + 2 + \lambda = 0$. It is orthogonal to $\mathrm { x } ( \mathrm { x } + 2 ) + ( \mathrm { y } + 1 ) ( \mathrm { y } - 3 ) = 0$ Or $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + 2 \mathrm { x } - 2 \mathrm { y } - 3 = 0$ so that $\frac { 2 ( 2 \lambda - 2 ) } { 2 } \cdot \left( \frac { 2 } { 2 } \right) + \frac { 2 ( 3 \lambda + 2 ) } { 2 } \left( \frac { - 2 } { 2 } \right) = 2 + \lambda - 3 \Rightarrow \lambda = - \frac { 3 } { 2 }$. Hence the required circle is $2 x ^ { 2 } + 2 y ^ { 2 } - 10 x - 5 y + 1 = 0$.

12. A circle touches the line $2 x + 3 y + 1 = 0$ at the point $( 1 , - 1 )$ and is orthogonal to the circle which has the line segment having end points $( 0 , - 1 )$ and $( - 2,3 )$ as the diameter.
Sol. Let the circle with tangent $2 \mathrm { x } + 3 \mathrm { y } + 1 = 0$ at $( 1 , - 1 )$ be $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + \lambda ( 2 x + 3 y + 1 ) = 0$ or $x ^ { 2 } + y ^ { 2 } + x ( 2 \lambda - 2 ) + y ( 3 \lambda + 2 ) + 2 + \lambda = 0$. It is orthogonal to $\mathrm { x } ( \mathrm { x } + 2 ) + ( \mathrm { y } + 1 ) ( \mathrm { y } - 3 ) = 0$ Or $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + 2 \mathrm { x } - 2 \mathrm { y } - 3 = 0$ so that $\frac { 2 ( 2 \lambda - 2 ) } { 2 } \cdot \left( \frac { 2 } { 2 } \right) + \frac { 2 ( 3 \lambda + 2 ) } { 2 } \left( \frac { - 2 } { 2 } \right) = 2 + \lambda - 3 \Rightarrow \lambda = - \frac { 3 } { 2 }$. Hence the required circle is $2 x ^ { 2 } + 2 y ^ { 2 } - 10 x - 5 y + 1 = 0$.

13. At any point $P$ on the parabola $y ^ { 2 } - 2 y - 4 x + 5 = 0$, a tangent is drawn which meets the directrix at $Q$. Find the locus of point R which divides QP externally in the ratio $\frac { 1 } { 2 } : 1$.
Sol. Any point on the parabola is $\mathrm { P } \left( 1 + \mathrm { t } ^ { 2 } , 1 + 2 \mathrm { t } \right)$. The equation of the tangent at P is $\mathrm { t } ( \mathrm { y } - 1 ) = \mathrm { x } - 1 + \mathrm { t } ^ { 2 }$ which meets the directrix $\mathrm { x } = 0$ at $\mathrm { Q } \left( 0,1 + \mathrm { t } - \frac { 1 } { \mathrm { t } } \right)$. Let R be $( \mathrm { h } , \mathrm { k } )$. Since it divides QP externally in the ratio $\frac { 1 } { 2 } : 1 , \mathrm { Q }$ is the mid point of RP $\Rightarrow 0 = \frac { \mathrm { h } + 1 + \mathrm { t } ^ { 2 } } { 2 }$ or $\mathrm { t } ^ { 2 } = - ( \mathrm { h } + 1 )$ and $1 + \mathrm { t } - \frac { 1 } { \mathrm { t } } = \frac { \mathrm { k } + 1 + 2 \mathrm { t } } { 2 }$ or $\mathrm { t } = \frac { 2 } { 1 - \mathrm { k } }$ So that $\frac { 4 } { ( 1 - k ) ^ { 2 } } + ( h + 1 ) = 0$ Or $( k - 1 ) ^ { 2 } ( h + 1 ) + 4 = 0$. Hence locus is $( y - 1 ) ^ { 2 } ( x + 1 ) + 4 = 0$.

13. At any point $P$ on the parabola $y ^ { 2 } - 2 y - 4 x + 5 = 0$, a tangent is drawn which meets the directrix at $Q$. Find the locus of point R which divides QP externally in the ratio $\frac { 1 } { 2 } : 1$.
Sol. Any point on the parabola is $\mathrm { P } \left( 1 + \mathrm { t } ^ { 2 } , 1 + 2 \mathrm { t } \right)$. The equation of the tangent at P is $\mathrm { t } ( \mathrm { y } - 1 ) = \mathrm { x } - 1 + \mathrm { t } ^ { 2 }$ which meets the directrix $\mathrm { x } = 0$ at $\mathrm { Q } \left( 0,1 + \mathrm { t } - \frac { 1 } { \mathrm { t } } \right)$. Let R be $( \mathrm { h } , \mathrm { k } )$. Since it divides QP externally in the ratio $\frac { 1 } { 2 } : 1 , \mathrm { Q }$ is the mid point of RP $\Rightarrow 0 = \frac { \mathrm { h } + 1 + \mathrm { t } ^ { 2 } } { 2 }$ or $\mathrm { t } ^ { 2 } = - ( \mathrm { h } + 1 )$ and $1 + \mathrm { t } - \frac { 1 } { \mathrm { t } } = \frac { \mathrm { k } + 1 + 2 \mathrm { t } } { 2 }$ or $\mathrm { t } = \frac { 2 } { 1 - \mathrm { k } }$ So that $\frac { 4 } { ( 1 - k ) ^ { 2 } } + ( h + 1 ) = 0$ Or $( k - 1 ) ^ { 2 } ( h + 1 ) + 4 = 0$. Hence locus is $( y - 1 ) ^ { 2 } ( x + 1 ) + 4 = 0$.

10. Tangents are drawn from any point on the hyperbola $x ^ { 2 } / 9 - y ^ { 2 } / 4 = 1$ to the circle $x ^ { 2 } + y ^ { 2 } = 9$. Find the locus of mid-point of the chord of contact.

11. Find the equation of the common tangent in 1st quadrant to the circle $x ^ { 2 } + y ^ { 2 } = 16$ and the ellipse $x ^ { 2 } / 25 + y ^ { 2 } / 4 = 1$. Also find the length of the intercept of the tangent between the coordinate axes.

11. Tangent at a point of the ellipse $x ^ { 2 } / a ^ { 2 } + y ^ { 2 } / b ^ { 2 } = 1$ is drawn which cuts the coordinate axes at $A$ and $B$. The minimum area of the triangle $O A B$ is ( $O$ being the origin) :
(a) $a b$
(b) $\left( a ^ { 3 } + a b + b ^ { 3 } \right) / 3$
(c) $a ^ { 2 } + b ^ { 2 }$
(d) $\left( \left( a ^ { 2 } + b ^ { 2 } \right) \right) / 4$

10. The axis of a parabola is along the line $\mathrm { y } = \mathrm { x }$ and the distance of its vertex from origin is $\sqrt { 2 }$ and that from its focus is $2 \sqrt { 2 }$. If vertex and focus both lie in the first quadrant, then the equation of the parabola is
(A) $( x + y ) ^ { 2 } = ( x - y - 2 )$
(B) $( x - y ) ^ { 2 } = ( x + y - 2 )$
(C) $( x - y ) ^ { 2 } = 4 ( x + y - 2 )$
(D) $( x - y ) ^ { 2 } = 8 ( x + y - 2 )$

Sol. (D)

Equation of directrix is $x + y = 0$. Hence equation of the parabola is
$$\frac { x + y } { \sqrt { 2 } } = \sqrt { ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } }$$
Hence equation of parabola is
$$( x - y ) ^ { 2 } = 8 ( x + y - 2 )$$

1. A plane passes through $( 1 , - 2,1 )$ and is perpendicular to two planes $2 x - 2 y + z = 0$ and $x - y + 2 z = 4$. The distance of the plane from the point $( 1,2,2 )$ is
   (A) 0
   (B) 1
   (C) $\sqrt { 2 }$
   (D) $2 \sqrt { 2 }$

Sol. (D) The plane is $\mathrm { a } ( \mathrm { x } - 1 ) + \mathrm { b } ( \mathrm { y } + 2 ) + \mathrm { c } ( \mathrm { z } - 1 ) = 0$ where $2 \mathrm { a } - 2 \mathrm {~b} + \mathrm { c } = 0$ and $\mathrm { a } - \mathrm { b } + 2 \mathrm { c } = 0$ $\Rightarrow \frac { \mathrm { a } } { 1 } = \frac { \mathrm { b } } { 1 } = \frac { \mathrm { c } } { 0 }$ So, the equation of plane is $x + y + 1 = 0$ ∴ $\quad$ Distance of the plane from the point $( 1,2,2 ) = \frac { 1 + 2 + 1 } { \sqrt { 1 ^ { 2 } + 1 ^ { 2 } } } = 2 \sqrt { 2 }$.

27. If P is a point on $\mathrm { C } _ { 1 }$ and Q in another point on $\mathrm { C } _ { 2 }$, then $\frac { \mathrm { PA } ^ { 2 } + \mathrm { PB } ^ { 2 } + \mathrm { PC } ^ { 2 } + \mathrm { PD } ^ { 2 } } { \mathrm { QA } ^ { 2 } + \mathrm { QB } ^ { 2 } + \mathrm { QC } ^ { 2 } + \mathrm { QD } ^ { 2 } }$ is equal to
(A) 0.75
(B) 1.25
(C) 1
(D) 0.5

Sol. (A)

Let $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D be the complex numbers $\sqrt { 2 } , - \sqrt { 2 } , \sqrt { 2 } \mathrm { i }$ and $- \sqrt { 2 } \mathrm { i }$ respectively. $\Rightarrow \frac { \mathrm { PA } ^ { 2 } + \mathrm { PB } ^ { 2 } + \mathrm { PC } ^ { 2 } + \mathrm { PD } ^ { 2 } } { \mathrm { QA } ^ { 2 } + \mathrm { QB } ^ { 2 } + \mathrm { QC } ^ { 2 } + \mathrm { QD } ^ { 2 } } = \frac { \left| \mathrm { z } _ { 1 } - \sqrt { 2 } \right| ^ { 2 } + \left| \mathrm { z } _ { 1 } + \sqrt { 2 } \right| ^ { 2 } + \left| \mathrm { z } _ { 1 } + \sqrt { 2 } \mathrm { i } \right| ^ { 2 } + \left| \mathrm { z } _ { 1 } - \sqrt { 2 } \mathrm { i } \right| ^ { 2 } } { \left| \mathrm { z } _ { 2 } + \sqrt { 2 } \right| ^ { 2 } + \left| \mathrm { z } _ { 2 } - \sqrt { 2 } \right| ^ { 2 } + \left| \mathrm { z } _ { 2 } - \sqrt { 2 } \mathrm { i } \right| ^ { 2 } + \left| \mathrm { z } _ { 2 } + \sqrt { 2 } \mathrm { i } \right| ^ { 2 } } = \frac { \left| \mathrm { z } _ { 1 } \right| ^ { 2 } + 2 } { \left| \mathrm { z } _ { 2 } \right| ^ { 2 } + 2 } = \frac { 3 } { 4 }$.

28. A circle touches the line L and the circle $\mathrm { C } _ { 1 }$ externally such that both the circles are on the same side of the line, then the locus of centre of the circle is
(A) ellipse
(B) hyperbola
(C) parabola
(D) parts of straight line
Sol. (C) Let C be the centre of the required circle. Now draw a line parallel to L at a distance of $\mathrm { r } _ { 1 }$ (radius of $\mathrm { C } _ { 1 }$ ) from it. Now $\mathrm { CP } _ { 1 } = \mathrm { AC } \Rightarrow \mathrm { C }$ lies on a parabola. [Figure]

29. A line M through A is drawn parallel to BD . Point S moves such that its distances from the line BD and the vertex A are equal. If locus of $S$ cuts $M$ at $T _ { 2 }$ and $T _ { 3 }$ and $A C$ at $T _ { 1 }$, then area of $\Delta T _ { 1 } T _ { 2 } T _ { 3 }$ is
(A) $\frac { 1 } { 2 }$ sq. units
(B) $\frac { 2 } { 3 }$ sq. units
(C) 1 sq. unit
(D) 2 sq. units
Sol. (C) $\because \mathrm { AG } = \sqrt { 2 }$ $\therefore \mathrm { AT } _ { 1 } = \mathrm { T } _ { 1 } \mathrm { G } = \frac { 1 } { \sqrt { 2 } } \quad \left[ \right.$ as A is the focus, $\mathrm { T } _ { 1 }$ is the vertex and BD is the directrix of parabola]. Also $\mathrm { T } _ { 2 } \mathrm {~T} _ { 3 }$ is latus rectum $\therefore \mathrm { T } _ { 2 } \mathrm {~T} _ { 3 } = 4 \times \frac { 1 } { \sqrt { 2 } }$ ∴ Area of $\Delta \mathrm { T } _ { 1 } \mathrm {~T} _ { 2 } \mathrm {~T} _ { 3 } = \frac { 1 } { 2 } \times \frac { 1 } { \sqrt { 2 } } \times \frac { 4 } { \sqrt { 2 } } = 1$. [Figure]

Comprehension IV

$A = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{array} \right]$, if $U _ { 1 } , U _ { 2 }$ and $U _ { 3 }$ are columns matrices satisfying. $\mathrm { AU } _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right] , \mathrm { AU } _ { 2 } = \left[ \begin{array} { l } 2 \\ 3 \\ 0 \end{array} \right] , \quad \mathrm { AU } _ { 3 } = \left[ \begin{array} { l } 2 \\ 3 \\ 1 \end{array} \right]$ and U is $3 \times 3$ matrix whose columns are $\mathrm { U } _ { 1 } , \mathrm { U } _ { 2 } , \mathrm { U } _ { 3 }$ then answer the following questions

Consider the two curves $$\begin{aligned} & C _ { 1 } : y ^ { 2 } = 4 x \\ & C _ { 2 } : x ^ { 2 } + y ^ { 2 } - 6 x + 1 = 0 \end{aligned}$$ Then,
(A) $C _ { 1 }$ and $C _ { 2 }$ touch each other only at one point
(B) $C _ { 1 }$ and $C _ { 2 }$ touch each other exactly at two points
(C) $C _ { 1 }$ and $C _ { 2 }$ intersect (but do not touch) at exactly two points
(D) $C _ { 1 }$ and $C _ { 2 }$ neither intersect nor touch each other

A straight line through the vertex $P$ of a triangle $P Q R$ intersects the side $Q R$ at the point $S$ and the circumcircle of the triangle $P Q R$ at the point $T$. If $S$ is not the centre of the circumcircle, then
(A) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 2 } { \sqrt { Q S \times S R } }$
(B) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 2 } { \sqrt { Q S \times S R } }$
(C) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 4 } { Q R }$
(D) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 4 } { Q R }$

Let $P \left( x _ { 1 } , y _ { 1 } \right)$ and $Q \left( x _ { 2 } , y _ { 2 } \right) , y _ { 1 } < 0 , y _ { 2 } < 0$, be the end points of the latus rectum of the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$. The equations of parabolas with latus rectum $P Q$ are
(A) $x ^ { 2 } + 2 \sqrt { 3 } \quad y = 3 + \sqrt { 3 }$
(B) $x ^ { 2 } - 2 \sqrt { 3 } \quad y = 3 + \sqrt { 3 }$
(C) $x ^ { 2 } + 2 \sqrt { 3 } \quad y = 3 - \sqrt { 3 }$
(D) $x ^ { 2 } - 2 \sqrt { 3 } \quad y = 3 - \sqrt { 3 }$

Consider
$$\begin{aligned} & L _ { 1 } : 2 x + 3 y + p - 3 = 0 \\ & L _ { 2 } : 2 x + 3 y + p + 3 = 0 \end{aligned}$$
where $p$ is a real number, and $C : x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 30 = 0$. STATEMENT-1 : If line $L _ { 1 }$ is a chord of circle $C$, then line $L _ { 2 }$ is not always a diameter of circle $C$.
and
STATEMENT-2 : If line $L _ { 1 }$ is a diameter of circle $C$, then line $L _ { 2 }$ is not a chord of circle $C$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
The equation of circle $C$ is
(A) $\quad ( x - 2 \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$
(B) $( x - 2 \sqrt { 3 } ) ^ { 2 } + \left( y + \frac { 1 } { 2 } \right) ^ { 2 } = 1$
(C) $\quad ( x - \sqrt { 3 } ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$
(D) $( x - \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
Points $E$ and $F$ are given by
(A) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right) , ( \sqrt { 3 } , 0 )$
(B) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right) , ( \sqrt { 3 } , 0 )$
(C) $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right) , \left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$
(D) $\left( \frac { 3 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right) , \left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right)$