Mathematics

Geometry and Trigonometry

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

  1. $\displaystyle \frac{\pi }{4}$
  2. $\displaystyle \frac{\pi }{3}$
  3. $\displaystyle \frac{\pi }{2}$

  4. none
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $y + 1 = m (x - 7) + \sqrt{25}(\sqrt{m^2 + 1})$ be any line to the circle.

Since we need tangents form $(0,0)$

$(0+1) = m(0 – 7) + 5\sqrt{m^2 + 1}$

$(7m + 1)^2 = 25(m^2 + 1)$

$\implies 24m^2 + 14m – 24 =0$

If $m _1, m _2$ are roots of the equation

$m _1m _2 = \dfrac{c}{a} = \dfrac{-24}{24} = -1$

Lines with $m _1$ and $m _2$ are slope are perpendicular.

Tangents from origin are at right angles to each other.

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 $\displaystyle 60^{\circ}$ are drawn to a circle of radius 3 cm then length of each tangent is equal to

  1. $\displaystyle \frac{3}{2}\sqrt{3}cm$
  2. $6 cm$
  3. $3 cm$
  4. $\displaystyle 3\sqrt{3}cm$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let PA and PB are the tangents on the circle. $\angle APB = 60$. the radius of the circle with center at O be 3 cm.
The two tangents drawn to a circle from an external point are equally inclined to the segment joining the center to the point.
Thus, $\angle APO = 30^{\circ}$
In $\triangle OAP$
$\angle OAP = 90^{\circ}$       ...(Angle between tangent and radius)
$\tan 30 = \cfrac{1}{\sqrt{3}} = \dfrac{OA}{AP}$
$PA = 3 \sqrt{3}$

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

If $5x-12y+10=0$ and $12y-5x+16=0$ are two tangents
to a circle then radius of the circle is

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

$5x-12y+10=0$ and $12y-5x+16=0$ are two parallel tangent to a circle.
Then distance $bet^{n}$ this two parallel tangents is $2r$.
$\therefore d=\left | \dfrac{-10-16}{\sqrt{5^{2}+12^{2}}} \right |=\left | \dfrac{26}{13} \right |=2$
$\therefore \ d=2r=2$
$\Rightarrow r=radius=1$

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

If ${ \theta } _{ 1 },{ \theta } _{ 2 }$ be the inclinations of tangents drawn from the point $P$ to the circle ${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$ and $\cot { { \theta  } _{ 1 } } +\cot { { \theta  } _{ 2 } } =k$, then the locus of $P$ is

  1. $k\left( { y }^{ 2 }+{ a }^{ 2 } \right) =2xy$
  2. $k\left( { y }^{ 2 }-{ a }^{ 2 } \right) =2xy$
  3. $k\left( { y }^{ 2 }+{ a }^{ 2 } \right) =4xy$

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

Equation of the circle is ${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$    ...(1)

Let $P$ be the point $\left( { x } _{ 1 },{ y } _{ 1 } \right) $.
Equation of any tangent to (1) is $y=mx+a\sqrt { 1+{ m }^{ 2 } } $
It is passes through $P\left( { x } _{ 1 },{ y } _{ 1 } \right) $, then
${ y } _{ 1 }=m{ x } _{ 1 }+a\sqrt { 1+{ m }^{ 2 } } \Rightarrow { y } _{ 1 }-m{ x } _{ 1 }=a\sqrt { 1+{ m }^{ 2 } } $
Squaring ${ { y } _{ 1 } }^{ 2 }+2mx _{ 1 }{ y } _{ 1 }+{ m }^{ 2 }{ { x } _{ 1 } }^{ 2 }={ a }^{ 2 }\left( 1+{ m }^{ 2 } \right)$
$ \Rightarrow \left( { { x } _{ 1 } }^{ 2 }-{ a }^{ 2 } \right) { m }^{ 2 }-2{ x } _{ 1 }{ y } _{ 1 }m+\left( { { y } _{ 1 } }^{ 2 }-{ a }^{ 2 } \right) =0$   ...(2)
This is a quadratic in $m$. If ${ m } _{ 1 }$ and ${ m } _{ 2 }$ are its roots, then these are the slopes of the tangents from $P$.
Since inclination of tangents are given to be ${\theta} _{1}$ and ${\theta} _{2}$
$\therefore$ Let ${ m } _{ 1 }=\tan{{\theta} _{1}}$ and ${ m } _{ 2 }=\tan{{\theta} _{2}}$ 
$\displaystyle \Rightarrow \frac { 1 }{ { m } _{ 1 } } +\frac { 1 }{ { m } _{ 2 } } =k\Rightarrow { m } _{ 1 }+{ m } _{ 2 }=k{ m } _{ 1 }{ m } _{ 2 }$
$\displaystyle \therefore \frac { 2{ x } _{ 1 }{ y } _{ 1 } }{ { { x } _{ 1 } }^{ 2 }-{ a }^{ 2 } } =k.\frac { { { y } _{ 1 } }^{ 2 }-{ a }^{ 2 } }{ { { x } _{ 1 } }^{ 2 }-{ a }^{ 2 } } \Rightarrow 2{ x } _{ 1 }{ y } _{ 1 }=k\left( { { y } _{ 1 } }^{ 2 }-{ a }^{ 2 } \right) $
$\therefore $ Locus of $P$ is $k\left( { y }^{ 2 }{ -a }^{ 2 } \right) =2xy$

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 tangents from the origin to the circle $(x-7)^{2}+(y+1)^{2}=25$ is

  1. $\displaystyle \frac{\pi}{3}$
  2. $\displaystyle \frac{\pi}{6}$
  3. $\displaystyle \frac{\pi}{2}$
  4. $\displaystyle \frac{\pi}{8}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(x-7)^2+(y+1)^2=25$
PA=PB=length of tangent from $(0,0) \space  to \space  (x-7)^2+(y+1)^2-25=0$
$=\sqrt{51}$
$\Rightarrow PA=PB=\sqrt{7^2+1-25}=5$
In $\Delta  OAP,$
$\tan  \alpha =\dfrac{OA}{PA}=\dfrac{5}{5}=1$
$\alpha =45^{\circ}$
So, angle both tangents $ =2\alpha =90^{\circ}$

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

State true or false
The length of tangent from an external point on a circle is always greater than the radius of the circle.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

false, it is not always required it can even be less or greater than the radius of the circle, it depend on how far the point is from the center of the circle. 

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

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

true 

since if the tangent intersects the circle at Q the PQO forms a right angled triangle with hypotenuse PO so the length PQ is always less than PO as hypotenuse is the largest in a triangle.

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

Two tangents are drawn to a circle and the angle between them is $\displaystyle { 30 }^{ \circ  }$. What is the angle between the radii that are drawn at the point of contact of these two tangents.

  1. $\displaystyle { 30 }^{ \circ }$
  2. $\displaystyle { 60 }^{ \circ }$
  3. $\displaystyle { 90 }^{ \circ }$
  4. $\displaystyle { 150 }^{ \circ }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The angle between the two tangents and the angle between the radii at the points of contact are supplementary, as they form a quadrilateral with two 90-degree angles. Thus, 180 - 30 = 150 degrees.

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. $