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Recent questions and answers in Geometry

0 votes
1 answer
1
Suppose the circle with equation $x^2+y^2+2fx+2gy+c=0$ cuts the parabola $y^2=4ax, \: (a>0)$ at four distinct points. If $d$ denotes the sum of the ordinates of these four points, then the set of possible values of $d$ is $\{0\}$ $(-4a,4a)$ $(-a,a)$ $(- \infty, \infty)$
answered May 18 in Geometry Amartya 129 views
0 votes
1 answer
2
Consider the family $\mathcal{F}$ of curves in the plane given by $x=cy^2$, where $c$ is a real parameter. Let $\mathcal{G}$ be the family of curves having the following property: every member of $\mathcal{G}$ intersect each member of $\mathcal{F}$ orthogonally. Then $\mathcal{G}$ is given by $xy=k$ $x^2+y^2=k^2$ $y^2+2x^2=k^2$ $x^2-y^2+2yk=k^2$
answered May 18 in Geometry Amartya 116 views
0 votes
1 answer
3
Let $A$ be the point of intersection of the lines $3x-y=1$ and $y=1$. Let $B$ be the point of reflection of the point $A$ with respect to the $y$-axis. Then the equation of the straight line through $B$ that produces a right angled triangle $ABC$ with $\angle ABC=90^{\circ}$, and $C$ lies on the line $3x-y=1$, is $3x-3y=2$ $2x+3=0$ $3x+2=0$ $3y-2=0$
answered Mar 15 in Geometry haralk10 57 views
0 votes
1 answer
4
If the distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2},$ then the equation of the hyperbola is $y^{2}-x^{2}=32$ $x^{2}-y^{2}=16$ $y^{2}-x^{2}=16$ $x^{2}-y^{2}=32$
answered Mar 15 in Geometry haralk10 45 views
0 votes
1 answer
5
If the tangent at the point $P$ with coordinates $(h,k)$ on the curve $y^2=2x^3$ is perpendicular to the straight line $4x=3y$, then $(h,k) = (0,0)$ $(h,k) = (1/8, -1/16)$ $(h,k) = (0,0) \text{ or } (h,k) = (1/8, -1/16)$ no such point $(h,k)$ exists
answered Mar 14 in Geometry haralk10 134 views
0 votes
1 answer
6
Four tangents are drawn to the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{5}=1$ at the ends of its latera recta. The area of the quadrilateral so formed is $27$ $\frac{13}{2}$ $\frac{15}{4}$ $45$
answered Mar 11 in Geometry haralk10 75 views
0 votes
1 answer
7
The area bounded by $y=x^{2}-4,y=0$ and $x=4$ is $\frac{64}{3}$ $6$ $\frac{16}{3}$ $\frac{32}{3}$
answered Mar 2 in Geometry haralk10 53 views
0 votes
1 answer
8
The area under the curve $x^2+3x-4$ in the positive quadrant and bounded by the line $x=5$ is equal to $59 \frac{1}{6}$ $61 \frac{1}{3}$ $40 \frac{2}{3}$ $72$
answered Mar 2 in Geometry haralk10 50 views
1 vote
2 answers
9
The value of $\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15^{\circ}+\cdots+\sin^{2}90^{\circ}$ is $8$ $9$ $9.5$ None of these
answered Feb 1 in Geometry Lakshman Patel RJIT 67 views
0 votes
1 answer
10
The area of the region bounded by the curves $y=\sqrt x,$ $2y+3=x$ and $x$-axis in the first quadrant is $9$ $\frac{27}{4}$ $36$ $18$
answered Nov 29, 2019 in Geometry techbd123 102 views
1 vote
1 answer
11
If $x+y=\pi, $ the expression $\cot \dfrac{x}{2}+\cot\dfrac{y}{2}$ can be written as $2 \: \text{cosec} \: x$ $\text{cosec} \: x + \text{cosec} \: y$ $2 \: \sin x$ $\sin x+\sin y$
answered Nov 28, 2019 in Geometry Lakshman Patel RJIT 52 views
2 votes
1 answer
12
The value of $\tan \left(\sin^{-1}\left(\frac{3}{5}\right)+\cot^{-1}\left(\frac{3}{2}\right)\right)$ is $\frac{1}{18}$ $\frac{11}{6}$ $\frac{13}{6}$ $\frac{17}{6}$
answered Nov 28, 2019 in Geometry Lakshman Patel RJIT 71 views
0 votes
1 answer
13
Let the sides opposite to the angles $A,B,C$ in a triangle $ABC$ be represented by $a,b,c$ respectively. If $(c+a+b)(a+b-c)=ab,$ then the angle $C$ is $\frac{\pi}{6}$ $\frac{\pi}{3}$ $\frac{\pi}{2}$ $\frac{2\pi}{3}$
answered Nov 21, 2019 in Geometry `JEET 51 views
0 votes
1 answer
14
The coordinates of a moving point $P$ satisfy the equations $\frac{dx}{dt} = \tan x, \:\:\:\: \frac{dy}{dt}=-\sin^2x, \:\:\:\:\: t \geq 0.$ If the curve passes through the point $(\pi/2, 0)$ when $t=0$, then the equation of the curve in rectangular co-ordinates is $y=1/2 \cos ^2 x$ $y=\sin 2x$ $y=\cos 2x+1$ $y=\sin ^2 x-1$
answered Nov 19, 2019 in Geometry `JEET 131 views
0 votes
1 answer
15
The area (in square unit) of the portion enclosed by the curve $\sqrt{2x}+ \sqrt{2y} = 2 \sqrt{3}$ and the axes of reference is $2$ $4$ $6$ $8$
answered Nov 18, 2019 in Geometry imShreyas 68 views
0 votes
1 answer
16
If $\sin^{-1}\frac{1}{\sqrt{5}}$ and $\cos^{-1}\frac{3}{\sqrt{10}}$ lie in $\left[0,\frac{\pi}{2}\right],$ their sum is equal to $\frac{\pi}{6}$ $\frac{\pi}{3}$ $\sin^{-1}\frac{1}{\sqrt{50}}$ $\frac{\pi}{4}$
answered Oct 20, 2019 in Geometry `JEET 74 views
0 votes
2 answers
17
If $\tan\: x=p+1$ and $\tan\; y=p-1,$ then the value of $2\:\cot\:(x-y)$ is $2p$ $p^{2}$ $(p+1)(p-1)$ $\frac{2p}{p^{2}-1}$
answered Oct 13, 2019 in Geometry techbd123 70 views
1 vote
1 answer
18
Let $g(x,y) = \text{max}\{12-x, 8-y\}$. Then the minimum value of $g(x,y)$ $ $ as $(x,y)$ varies over the line $x+y =10$ is $5$ $7$ $1$ $3$
answered Oct 4, 2019 in Geometry `JEET 81 views
0 votes
1 answer
19
The value of $\sin^{6}\frac{\pi}{81}+\cos^{6}\frac{\pi}{81}-1+3\sin^{2}\frac{\pi}{81}\:\cos^{2}\frac{\pi}{81}$ is $\tan^{6}\frac{\pi}{81}$ $0$ $-1$ None of these
answered Oct 2, 2019 in Geometry `JEET 47 views
0 votes
1 answer
20
If $\cos2\theta=\sqrt{2}(\cos\theta-\sin\theta)$ then $\tan\theta$ equals $1$ $1$ or $-1$ $\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}$ or $1$ None of these
answered Oct 2, 2019 in Geometry `JEET 53 views
0 votes
1 answer
21
If $A(t)$ is the area of the region bounded by the curve $y=e^{-\mid x \mid}$ and the portion of the $x$-axis between $-t$ and $t$, then $\underset{t \to \infty}{\lim} A(t)$ equals $0$ $1$ $2$ $4$
answered Oct 1, 2019 in Geometry techbd123 72 views
0 votes
1 answer
22
If $\sin(\sin^{-1}\frac{2}{5}+\cos^{-1}x)=1,$ then $x$ is $1$ $\frac{2}{5}$ $\frac{3}{5}$ None of these
answered Sep 30, 2019 in Geometry `JEET 42 views
0 votes
1 answer
23
The medians $AD$ and $BE$ of the triangle with vertices $A(0,b),B(0,0)$ and $C(a,0)$ are mutually perpendicular if $b=\sqrt{2}a$ $a=\pm\sqrt{2}b$ $b=-\sqrt{2}a$ $b=a$
answered Sep 27, 2019 in Geometry `JEET 49 views
0 votes
1 answer
24
The polar equation $r=a \cos \theta$ represents a spiral a parabola a circle none of the above
answered Sep 24, 2019 in Geometry `JEET 103 views
0 votes
1 answer
25
If a square of side $a$ and an equilateral triangle of side $b$ are inscribed in a circle then $a/b$ equals $\sqrt{2/3}$ $\sqrt{3/2}$ $3/ \sqrt{2}$ $\sqrt{2}/3$
answered Sep 24, 2019 in Geometry `JEET 127 views
0 votes
0 answers
26
Let $y^2-4ax+4a=0$ and $x^2+y^2-2(1+a)x+1+2a-3a^2=0$ be two curves. State which one of the following statements is true. These two curves intersect at two points These two curves are tangent to each other These two curves intersect orthogonally at one point These two curves do not intersect
asked Sep 23, 2019 in Geometry Arjun 70 views
0 votes
0 answers
27
If $f(x)=x^2$ and $g(x)= x \sin x + \cos x$ then $f$ and $g$ agree at no points $f$ and $g$ agree at exactly one point $f$ and $g$ agree at exactly two points $f$ and $g$ agree at more than two points
asked Sep 23, 2019 in Geometry Arjun 128 views
0 votes
0 answers
28
Angles between any pair of $4$ main diagonals of a cube are $\cos^{-1} 1/\sqrt{3}, \pi – \cos ^{-1} 1/\sqrt{3}$ $\cos^{-1} 1/3, \pi – \cos ^{-1} 1/3$ $\pi/2$ none of the above
asked Sep 23, 2019 in Geometry Arjun 102 views
0 votes
0 answers
29
Let the position of a particle in three dimensional space at time $t$ be $(t, \cos t, \sin t)$. Then the length of the path traversed by the particle between the times $t=0$ and $t=2 \pi$ is $2 \pi$ $2 \sqrt{2 \pi}$ $\sqrt{2 \pi}$ none of the above
asked Sep 23, 2019 in Geometry Arjun 85 views
0 votes
0 answers
30
The length of the curve $x=t^3$, $y=3t^2$ from $t=0$ to $t=4$ is $5 \sqrt{5}+1$ $8(5 \sqrt{5}+1)$ $5 \sqrt{5}-1$ $8(5 \sqrt{5}-1)$
asked Sep 23, 2019 in Geometry Arjun 96 views
0 votes
0 answers
31
The volume of the solid, generated by revolving about the horizontal line $y=2$ the region bounded by $y^2 \leq 2x$, $x \leq 8$ and $y \geq 2$, is $2 \sqrt{2\pi}$ $28 \pi/3$ $84 \pi$ none of the above
asked Sep 23, 2019 in Geometry Arjun 79 views
0 votes
1 answer
32
The set $\{(x,y)\: :\: \mid x\mid+\mid y\mid\:\leq\:1\}$ is represented by the shaded region in
answered Sep 19, 2019 in Geometry Bharat Makhija 50 views
0 votes
1 answer
33
There are three circles of equal diameter ($10$ units each) as shown in the figure below. The straight line $PQ$ passes through the centres of all the three circles. The straight line $PR$ is a tangent to the third circle at $C$ and cuts the second circle at the points $A$ and $B$ as shown in the figure.Then the length of the line segment $AB$ is $6$ units $7$ units $8$ units $9$ units
answered Sep 19, 2019 in Geometry ankitgupta.1729 81 views
0 votes
0 answers
34
The shaded region in the following diagram represents the relation $y\:\leq\: x$ $\mid \:y\mid \:\leq\: \mid x\:\mid $ $y\:\leq\: \mid x\:\mid$ $\mid \:y\mid\: \leq\: x$
asked Sep 18, 2019 in Geometry gatecse 46 views
0 votes
0 answers
35
The length of the chord on the straight line $3x-4y+5=0$ intercepted by the circle passing through the points $(1,2),(3,-4)$ and $(5,6)$ is $12$ $14$ $16$ $18$
asked Sep 18, 2019 in Geometry gatecse 38 views
0 votes
0 answers
36
The equations $x=a\cos\theta+b\sin\theta$ and $y=a\sin\theta+b\cos\theta,(0\leq\theta\leq2\pi$ and $a,b$ are arbitrary constants$)$ represent a circle a parabola an ellipse a hyperbola
asked Sep 18, 2019 in Geometry gatecse 45 views
0 votes
0 answers
37
A straight line touches the circle $x^{2}+y^{2}=2a^{2}$ and also the parabola $y^{2}=8ax.$ Then the equation of the straight line is $y=\pm x$ $y=\pm(x+a)$ $y=\pm(x+2a)$ $y=\pm(x-21)$
asked Sep 18, 2019 in Geometry gatecse 32 views
0 votes
0 answers
38
In an ellipse, the distance between its foci is $6$ and its minor axis is $8.$ Then its eccentricity is $\frac{4}{5}$ $\frac{1}{\sqrt{52}}$ $\frac{3}{5}$ $\frac{1}{2}$
asked Sep 18, 2019 in Geometry gatecse 33 views
0 votes
0 answers
39
If in a $\triangle ABC,\angle B=\dfrac{2\pi}{3},$ then $\cos A+\cos C$ lies in $\left[\:-\sqrt{3},\sqrt{3}\:\right]$ $\left(\:-\sqrt{3},\sqrt{3}\:\right]$ $\left(\:\frac{3}{2},\sqrt{3}\:\right)$ $\left(\:\frac{3}{2},\sqrt{3}\:\right]$
asked Sep 18, 2019 in Geometry gatecse 42 views
0 votes
0 answers
40
Which of the following relations is true for the following figure? $b^{2}=c(c+a)$ $c^{2}=a(a+b)$ $a^{2}=b(b+c)$ All of these
asked Sep 18, 2019 in Geometry gatecse 68 views
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