Mathematics

Geometry and Trigonometry

94 Questions

Geometry and trigonometry questions cover circles, tangents, and trigonometric ratios. Problems often combine algebraic geometry with angle properties to test spatial reasoning. This forms a core component of the mathematics section in engineering and civil services exams.

Circle tangentsTrigonometric ratiosHyperbola propertiesEllipse tangentsSecant construction

Geometry and Trigonometry Questions

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

The value of $k$ for which two tangents can be drawn from $(k , k)$ to the circle $x^2 + y^2 + 2x + 2y 16 = 0$ is

  1. $k\ \epsilon\ R^+$
  2. $k\ \epsilon \ R$
  3. $k\ \epsilon\ ( -\infty , -4) \cup ( 2, \infty )$
  4. $k\ \epsilon\ ( 0, 1]$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For two tangents to be drawn to $C(x, y):$

$\implies x^2 + y^2 +2x +2y – 16 = 0$

From P(k,k) , the point  P must lie outside the circle

$\implies C(k,k) > 0$

$\implies k^2 + k^2 + 2k + 2k – 16  > 0$

$\implies k^2 + 2k – 8 > 0$

$\implies (k + 4)(k - 2) > 0$

$\implies k \in (-\infty,-4) \cup (2,\infty)$

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

The area of the triangle formed by the tangents from the point $( 4, 3 )$ to the circle $x^{2} + y^{2} = 9$ and the line joining their points of contact is

  1. $\dfrac{25}{192}$ sq. units
  2. $\dfrac{192}{25}$ sq. units
  3. $\dfrac{384}{25}$ sq. units

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Area of triangle $ = \dfrac {RL^3}{L^2 + R^2}$
where R = radius of the circle = 3
L = length of tangent $ = \sqrt {S _1} = \sqrt {16 + 9 - 9} = 4$
Hence area $ = \dfrac {192}{25} sq.$ units

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

The angle between the two tangents from the origin to the circle ${ \left( x-7 \right)  }^{ 2 }+{ \left( y+1 \right)  }^{ 2 }=25$ equals

  1. $\cfrac { \pi }{ 6 } $
  2. $\cfrac { \pi }{ 3 } $
  3. $\cfrac { \pi }{ 2 } $
  4. $\cfrac { \pi }{ 4 } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation of any line through the origin $(0,0)$ is

$y = mx +c$

If it is a tangent to the circle $(x−7)2+(y+1)2=52$, then

 

$ \dfrac{\left| 7m+1 \right|}{\left| \sqrt{{{m}^{2}}+1} \right|}=5 $

$ {{\left( 7m+1 \right)}^{2}}=5.\left( {{m}^{2}}+1 \right) $

$ 24{{m}^{2}}+14m-24=0 $

 

This equation, being a quadratic in m, gives two values of m, say ${{m} _{1}}$  and${{m} _{2}}$  These two values of m are the slopes of the tangents drawn from the origin to the given circle.

From (i), we have ${{m} _{1}}\times {{m} _{2}}=-1$

 

Hence, the two tangents are perpendicular.

 

Ans: C

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

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

  1. $5$
  2. $6$
  3. $8$
  4. $4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the angle between the tangents is $\theta\implies \tan \dfrac{\theta}{2}=\dfrac{r}{\sqrt{6^{2}+8^{2}-r^{2}}}$

$\implies \sin \dfrac{\theta}{2}=\dfrac{r}{10},\cos \dfrac{\theta}{2}=\dfrac{\sqrt{100-r^{2}}}{10}$
Area of triangle is $\dfrac{1}{2}(\sqrt{S _{11}})^{2}\sin \theta=\dfrac{r(100-r^{2})^{3/2}}{100}$
Let $f(x)=r(100-r^{2})^{3/2}\implies f'(x)=(100-r^{2})^{1/2}(100-4{r^{2})}=0\implies r^{2}=25\implies r=5$
For area to be maximum $r=5$

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

Write True or False and justify your answer in each of the following :


The length of tangent from an external point P on a circle with centre O is always less than OP.

  1. True

  2. False

  3. Data insufficient

  4. Ambiguous
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$ Given-\ PA\quad &amp; \quad PB\quad are\quad tangents\quad to\quad the\quad circle\quad wtth\quad centre\quad O\ at\quad A\quad &amp; \quad B\quad respectively.\ To\quad find\quad out-\ The\quad assertion,\quad PA\quad or\quad PB\quad is\quad always\quad >\quad OA\quad or\quad OB,\quad is\quad \ true\quad or\quad false.\ Justification-\ PA\quad &amp; \quad PB\quad are\quad tangents\quad to\quad the\quad circle\quad at\quad A\quad &amp; \quad B\quad respectively.\ \therefore \quad PA\quad =\quad PB\quad and\quad \angle OAP\quad &amp; \quad \angle OBP\quad are={ 90 }^{ o }\ \therefore \quad \Delta OAP\quad is\quad a\quad right\quad one\quad with\quad \angle A={ 90 }^{ o }\ \Longrightarrow \angle AOP+\angle APO={ 90 }^{ o }\quad (by\quad angle\quad sum\quad prqperty\quad of\quad triangles)\ case\quad I-\quad \angle AOP=\angle APO\Longrightarrow each\quad of\quad them={ 45 }^{ o }.\quad i.e\quad PA=OA\ case\quad II-\quad \angle AOP>\angle APO\Longrightarrow \angle AOP>{ 45 }^{ o }\quad &amp; \quad \angle APO<{ 45 }^{ o }\quad \ (in\quad a\quad \Delta \quad the\quad side\quad opposite\quad to\quad the\quad greater\quad angle\quad is\quad greater\quad than\quad the\quad side\quad \ opposite\quad to\quad the\quad smaller\quad angle)\ \Longrightarrow PA>OA\ case\quad III-\quad \angle AOP<\angle APO\Longrightarrow \angle AOP{ <45 }^{ o }\quad &amp; \quad \angle APO>{ 45 }^{ o }\quad (\quad same\quad argument\quad as\quad case\quad II)\ \Longrightarrow PA>OA\ So\quad the\quad assertion,\quad PA\quad or\quad PB\quad is\quad always\quad <\quad OA\quad or\quad OB,\quad is\quad false.\ Ans-\quad False. $

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

To draw a pair of tangents to a circle which are inclined to each other at an angle of $60^0$, it is required to draw tangents at endpoints of those two radii of the circle, the angle between them should be

  1. $135^{0}$
  2. $90^{0}$
  3. $60^{0}$
  4. $120^{0}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given:

$ PA$ & $PB$ are two tangents drawn from P to a circle with centre O at A & B respectively. 
$\angle APB={ 60 }^{ o }$ 
To find out-
$ \angle AOB$ 
Solution-
$ PA$ & $PB$ are two tangents drawn from P to the circle at A & B, respectively.
$ \therefore  PA=PB\Longrightarrow \Delta PAB$ is isosceles. 
i.e $\angle PAB=\angle PBA.$ 
or $\angle PAB+\angle PBA=2\angle PAB$  .....(i)
Now, $\angle PAB+\angle PBA+\angle APB={ 180 }^{ o }$    ....(angle sum property of triangles)
$ \Longrightarrow \angle PAB+\angle PBA{ +60 }^{ o }={ 180 }^{ O }$
$\Longrightarrow 2\angle PAB={ 120 }^{ o }$     ...(from i)
$ \therefore  \angle PAB={ 60 }^{ o }=\angle PBA$  .........(ii)
Again, $\angle OAP={ 90 }^{ o }$    ....(angle between a radius and the tangent at the point of contact.)
$ \therefore \angle OAB=\angle OAP-\angle PAB={ 90 }^{ o }-{ 60 }^{ o }$    .... (from ii)    .........(iii)
Since, $OA=OB$     ...(radii of the same circle)
$ \therefore  \Delta OAB$ is isosceles
$\Longrightarrow \angle OAB=\angle OBA={ 30 }^{ o }$   ....(from iii)
So, $\angle AOB={ 180 }^{ o }-(\angle OAB+\angle OBA)={ 180 }^{ o }-{ 30 }^{ o }-{ 30 }^{ O }={ 120 }^{ O }$      ...(angle sum property of triangles)

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

If two tangents inclined at an angle of $60^{\circ}$ are drawn to a circle of radius 3 cm, then length of the tangent is equal to :

  1. $\sqrt{3}cm$
  2. $2\sqrt{3}cm$
  3. $\frac{2}{\sqrt{3}}cm$
  4. $3\sqrt{3}cm$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Tangent is perpendicular to radius at the point of contact.

By symmetry with respect to the line joining the center and the point from which tangents are drawn, we have the length of tangent $= \dfrac{3}{\tan 30^o} = 3\sqrt{3} cm$.
So option D is the right answer.

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

A tangent drawn from the point (4, 0) to the circle $\displaystyle x^{2}+y^{2}=8 $ touches it at a point A in the first quadrant. The coordinates of another point B on the circle such that $AB$ = 4 are

  1. $(2, -2)$
  2. $(-2, 2)$
  3. $\displaystyle \left ( -2\sqrt{2},0 \right ) $
  4. $\displaystyle \left ( 0,-2\sqrt{2} \right ) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x^2  + y^2 = (2\sqrt 2)^2 , C = (0,0) , r = 2 \sqrt 2$

Let $y = mx + 2\sqrt 2 \sqrt{m^2 + 1}$ be a tangent

To find the tangent through $(4,0)$ substitute into the equation

$\implies 0 = 4m +2\sqrt 2 \sqrt{m^2 + 1}$

$\implies 16m^2 = 8(m^2 + 1)$

$\implies m = -1$

Equation is

$y = -x + 4$

Substituting in circle equation

$x^2 + (-x + 4)^2 = 8$

$\implies 2x^2 – 8x + 8 = 0$

$\implies x = 2 \implies y = 2$

$A = (2,2)$

Any point on the circle is given be$ (2\sqrt2 \cos \theta, 2\sqrt2 \sin \theta )$

Let B = $(2\sqrt2 \cos \theta _1, 2\sqrt2 \sin \theta _1)$

$AB = 4 \implies AB^2 = 16$

$\implies (2\sqrt2 \cos \theta _1, -2)^2 + (2\sqrt2 \sin \theta _1 - 2)^2 = 16$

$\implies 8 + 4 + 4 – 4\sqrt 2(\cos \theta _1 + \sin \theta _1) = 16$

$\implies \sin \theta _1 + \cos \theta _1 = 0$

$\theta _1 = \dfrac{-\pi}{4}$

$B = (2\sqrt 2 \times\dfrac{1}{\sqrt 2} 2\sqrt 2 \times\dfrac{-1}{\sqrt 2})  = (2,-2)$

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

From a point $R(5, 8)$ two tangents $RP$ and $RQ$ are drawn to a given cirlce $S = 0$ whose radius is $5$. If circumcentre of the triangle PQR is $(2, 3)$, then the equation of circle $S= 0$ is

  1. $x^2 + y^2 + 2x + 4y - 20 = 0$
  2. $x^2 + y^2 + x + 2y - 10 = 0$
  3. $x^2 + y^2 - x - 2y - 20 = 0$
  4. $x^2 + y^2 - 4x - 6y - 12 = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The circumcenter of triangle PQR, where P and Q are points of tangency from R, is the midpoint of the chord of contact and the center of the circle. Using the given circumcenter and radius, the equation can be derived.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

A tangent to the circle $x^{2}+y^{2}=a^{2}$ meets the axes at points A and B. The locus of the mid point of AB is 

  1. $\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{1}{a^{2}}$
  2. $\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{4}{a^{2}}$
  3. $\frac{1}{x^{2}}+\frac{1}{y^{2}}=4a^{2}$
  4. $\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{a^{2}}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths ellipse normal to an ellipse tangent and normal to an ellipse two dimensional analytical geometry-ii

The number of tangents to the circle ${x}^{2}+{y}^{2}=3$ that are normals to the ellipse $\cfrac{{x}^{2}}{9}+\cfrac{{y}^{2}}{4}$ is

  1. one

  2. two

  3. three

  4. zero
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Let $y=mx+c$ is tangent to $x^2+y^2=3$ 
Then by condition of tangency
$\left|\dfrac{c}{\sqrt{m^2+1}}\right|=\sqrt{3}$
$\Rightarrow c^2=3(m^2+1)$        ...(i)
$y=mx+c$ is Normal to $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$
in $c=\dfrac{(b^2-a^2)m}{\sqrt{a^2+b^2m^2}}$

$\Rightarrow c=\dfrac{(4-9)m}{\sqrt{9+4m^2}}$

$\Rightarrow c^2=\dfrac{25m^2}{4m^2+9}$

$\Rightarrow 3(m^2+1)(4m^2+9)=25m^2$
let $m^2=t$
$\Rightarrow 3(t+1)(4t+9)=25t$
$\Rightarrow$ since, $D<0$.
Hence, There is no rout:
Hence, there is no tangent to circle which is Normal to ellipse.
Multiple choice maths ellipse normal to an ellipse tangent and normal to an ellipse two dimensional analytical geometry-ii

Tangents are drawn to the ellipse $ \displaystyle \frac{x^2}{a^2}+\displaystyle \frac{y^2}{b^2}=1 $ at points where it is intersected by the line $ \ell x+my+n=0 $. Find the point of intersection of tangents at these points.

  1. $ \displaystyle \frac{-a^2}{n},\displaystyle \frac{-b^2m}{n} $
  2. $ \displaystyle \frac{-a^2\ell}{n},\displaystyle \frac{-b^2m}{n} $
  3. $ \displaystyle \frac{-a^2\ell}{n},\displaystyle \frac{-b^2}{n} $

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $P\left( { x } _{ 1 },{ y } _{ 1 } \right) $ be the point of intersection of the
Line $lx+my+n=0$ and the ellipse $\cfrac { { x }^{ 2 }

}{ {a }^{ 2 } } +\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$
Then the equation of tangent at $P$ is
$\cfrac

{ { xx } _{ 1 } }{ { a }^{ 2 } } +\cfrac { { yy } _{ 1 } }{ { b }^{ 2 } }

=1\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot (i)$
Since $\left( { x } _{ 1 },{ y } _{ 1 } \right) $  is the point of intersection of the line
$lx+my+n=0\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot (ii)$
Clearly $(i)$ and $(ii)$ represent the same line. Therefore,
$\therefore \quad \cfrac { { x } _{ 1 } }{ { a }^{ 2 }l } =\cfrac { { y } _{ 1 } }{ { b }^{ 2 }m } =\cfrac { 1 }{ -n } $
${ x } _{ 1 }=\cfrac { { -a }^{ 2 }l }{ n } ,\quad { y } _{ 1 }=\cfrac { -{ b }^{ 2 }m }{ n } $
Therefore, the point of intersection of given line and  the given ellipse is
$\left(- \cfrac { { a }^{ 2 }l }{ n } ,\cfrac { { b }^{ 2 }m }{ n }  \right) \quad $
Hence, option 'B' is correct.

Multiple choice position of point wrt ellipse ellipse maths

A tangent to the ellipse $4x^2+9y^2=36$ is cut by tangent at the extremities of the major axis at $T$ and $T'$. The circles on $TT'$ as diameters passes through the point 

  1. $(0,-\sqrt5)$
  2. $(\sqrt5,0)$
  3. $(0,0)$
  4. $(3,2)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The ellipse 4x^2 + 9y^2 = 36 is x^2/9 + y^2/4 = 1. The tangents at the extremities of the major axis are x = 3 and x = -3. A tangent to the ellipse is y = mx + sqrt(9m^2 + 4). The intersection points T and T' with x=3 and x=-3 are found, and the circle with diameter TT' is constructed. This circle passes through the foci of the ellipse, which are at (+/- sqrt(9-4), 0) = (+/- sqrt(5), 0).

Multiple choice

What is a tangent space at a point on a manifold?

  1. The set of all tangent vectors at that point

  2. The set of all normal vectors at that point

  3. The set of all curves passing through that point

  4. The set of all surfaces passing through that point
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The tangent space at a point on a manifold is the vector space of all tangent vectors to the manifold at that point.