The question requires computing areas of regions bounded by circular arcs, areas of disks, areas of triangles/polygons inscribed in or circumscribed about circles, or ratios of such areas.
In the adjoining figure, $C$ is the centre of the circle drawn, $A, F, E$ lie on the circle and $BCDF$ is a rectangle. If $\frac{DE}{AB} = 2$, then $\frac{FE}{FA}$ equals (A) $\sqrt{\frac{3}{2}}$ (B) $\sqrt{2}$ (C) $\sqrt{\frac{5}{2}}$ (D) $\sqrt{3}$
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.
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
The common tangents to the circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$. Then the area of the quadrilateral $PQRS$ is (A) 3 (B) 6 (C) 9 (D) 15
Consider the ellipse $$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$$ Let $H ( \alpha , 0 ) , 0 < \alpha < 2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis. List-I (I) If $\phi = \frac { \pi } { 4 }$, then the area of the triangle $F G H$ is (II) If $\phi = \frac { \pi } { 3 }$, then the area of the triangle $F G H$ is (III) If $\phi = \frac { \pi } { 6 }$, then the area of the triangle $F G H$ is (IV) If $\phi = \frac { \pi } { 12 }$, then the area of the triangle $F G H$ is List-II (P) $\frac { ( \sqrt { 3 } - 1 ) ^ { 4 } } { 8 }$ (Q) 1 (R) $\frac { 3 } { 4 }$ (S) $\frac { 1 } { 2 \sqrt { 3 } }$ (T) $\frac { 3 \sqrt { 3 } } { 2 }$ The correct option is: (A) (I) → (R); (II) → (S); (III) → (Q); (IV) → (P) (B) (I) → (R); (II) → (T); (III) → (S); (IV) → (P) (C) (I) → (Q); (II) → (T); (III) → (S); (IV) → (P) (D) (I) → (Q); (II) → (S); (III) → (Q); (IV) → (P)
The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$, is (1) 27 (2) $\frac { 27 } { 4 }$ (3) 18 (4) $\frac { 27 } { 2 }$
Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac { 3 } { 5 }$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is: (1) 32 (2) 80 (3) 40 (4) 8
Let $S$ and $S ^ { \prime }$ be the foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta S ^ { \prime } B S$ is a right angled triangle with right angle at $B$ and area $\left( \Delta S ^ { \prime } B S \right) = 8$ sq. units, then the length of a latus rectum of the ellipse is : (1) $2 \sqrt { 2 }$ (2) 2 (3) 4 (4) $4 \sqrt { 2 }$
The tangent and the normal lines at the point $( \sqrt { 3 } , 1 )$ to the circle $x ^ { 2 } + y ^ { 2 } = 4$ and the $x$-axis form a triangle. The area of this triangle (in square units) is: (1) $\frac { 1 } { 3 }$ (2) $\frac { 2 } { \sqrt { 3 } }$ (3) $\frac { 4 } { \sqrt { 3 } }$ (4) $\frac { 1 } { \sqrt { 3 } }$
In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $( 0 , 5 \sqrt { 3 } )$, then the length of its latus rectum is: (1) 6 (2) 10 (3) 8 (4) 5
A rectangle is inscribed in a circle with a diameter lying along the line $3 y = x + 7$. If the two adjacent vertices of the rectangle are $( - 8,5 )$ and $( 6,5 )$, then the area of the rectangle (in sq. units) is: (1) 72 (2) 98 (3) 56 (4) 84
The area (in sq. units) of an equilateral triangle inscribed in the parabola $y ^ { 2 } = 8 x$, with one of its vertices on the vertex of this parabola is (1) $64 \sqrt { 3 }$ (2) $256 \sqrt { 3 }$ (3) $192 \sqrt { 3 }$ (4) $128 \sqrt { 3 }$
Two tangents are drawn from a point $P$ to the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y + 4 = 0$, such that the angle between these tangents is $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right)$, where $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right) \in ( 0 , \pi )$. If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$, then the ratio of the areas of $\triangle P A B$ and $\triangle C A B$ is: (1) $11 : 4$ (2) $9 : 4$ (3) $3 : 1$ (4) $2 : 1$
If the tangents drawn at the point $O(0,0)$ and $P(1+\sqrt{5}, 2)$ on the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at the point $Q$, then the area of the triangle $OPQ$ is equal to (1) $\frac{3+\sqrt{5}}{2}$ (2) $\frac{4+2\sqrt{5}}{2}$ (3) $\frac{5+3\sqrt{5}}{2}$ (4) $\frac{7+3\sqrt{5}}{2}$
Let the tangent to the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 2$ at the point $M ( - 1,1 )$ intersect the circle $C _ { 2 }$ : $( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 5$, at two distinct points $A$ and $B$. If the tangents to $C _ { 2 }$ at the points $A$ and $B$ intersect at $N$, then the area of the triangle $A N B$ is equal to (1) $\frac { 1 } { 2 }$ (2) $\frac { 2 } { 3 }$ (3) $\frac { 1 } { 6 }$ (4) $\frac { 5 } { 3 }$
The tangents at the points $A(1,3)$ and $B(1,-1)$ on the parabola $y^2 - 2x - 2y = 1$ meet at the point $P$. Then the area (in unit$^2$) of the triangle $PAB$ is: (1) 4 (2) 6 (3) 7 (4) 8
Let $C$ be the centre of the circle $x^2 + y^2 - x + 2y = \frac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $C$, makes an angle of $\frac{\pi}{4}$ with the line $CP$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $PQR$ (in unit$^2$) is (1) 2 (2) $2\sqrt{2}$ (3) $8\sin\frac{\pi}{8}$ (4) $8\cos\frac{\pi}{8}$
A point $P$ moves so that the sum of squares of its distances from the points $( 1,2 )$ and $( - 2,1 )$ is 14. Let $f ( x , y ) = 0$ be the locus of $P$, which intersects the $x$-axis at the points $A , B$ and the $y$-axis at the point $C , D$. Then the area of the quadrilateral $ACBD$ is equal to (1) $\frac { 9 } { 2 }$ (2) $\frac { 3 \sqrt { 17 } } { 2 }$ (3) $\frac { 3 \sqrt { 17 } } { 4 }$ (4) 9
If the tangents at the points $P$ and $Q$ on the circle $x^{2} + y^{2} - 2x + y = 5$ meet at the point $R\left(\frac{9}{4}, 2\right)$, then the area of the triangle $PQR$ is (1) $\frac{5}{4}$ (2) $\frac{13}{8}$ (3) $\frac{5}{8}$ (4) $\frac{13}{4}$
Let $P(a_{1}, b_{1})$ and $Q(a_{2}, b_{2})$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$, then $a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}$ is equal to $\_\_\_\_$
Let a circle $C_1$ be obtained on rolling the circle $x^2 + y^2 - 4x - 6y + 11 = 0$ upwards 4 units on the tangent $T$ to it at the point $(3,2)$. Let $C_2$ be the image of $C_1$ in $T$. Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is: (1) $22 + \sqrt{2}$ (2) $41 + \sqrt{2}$ (3) $3 + 2\sqrt{2}$ (4) $21 + \sqrt{2}$
Let the ellipse $E$: $x^2 + 9y^2 = 9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A$, $P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m - n$ is equal to (1) 16 (2) 15 (3) 17 (4) 18
The area (in sq. units) of the part of circle $x ^ { 2 } + y ^ { 2 } = 169$ which is below the line $5 x - y = 13$ is $\frac { \pi \alpha } { 2 \beta } - \frac { 65 } { 2 } + \frac { \alpha } { \beta } \sin ^ { - 1 } \left( \frac { 12 } { 13 } \right)$ where $\alpha , \beta$ are coprime numbers. Then $\alpha + \beta$ is equal to