Quantitative Aptitude

Geometry

1,950 Questions
All Circle Tangents and Chords 772 Coordinate Geometry 360 Circle and Arc Measures 216 Triangle Centers and Medians 175 Composite and Inscribed Shapes 174 Similarity, Congruence, and Proportionality 118 Right Triangles and Pythagorean Theorem 100 Standard 2D Mensuration 22 Polygon and Angle Sums 10 3D Solid Mensuration 3

Geometry Questions

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 A which is at a distance of 10 cm from the center O of a circle of radius 6 cm, the pair of tangents AB and AC to the circle are drawn. Then the area of Quadrilateral ABOC is:

  1. $24 cm^{2}$
  2. $4 8cm^{2}$
  3. $96cm^{2}$
  4. $100cm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since $\triangle ABO$ is congruent to $\triangle ACO$,  area of $ABOC$ is twice the area of $\triangle ABO$.

In $\triangle ABO, \ OA = 10 cm, \ OB = 6 cm$.
Since tangent is perpendicular to radius at the point of contact, by Pythagoras' theorem, we have 
$AB = \sqrt{OA^2 - OB^2} = 8 cm$
So, the area of $\triangle ABO$ is $\dfrac{1}{2}\times AB\times OB = 24 cm^2$
So, the area of $ABOC$ is $2\times 24 = 48 cm^2$.
So option B 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

If the angle between two radii of a circle is $140^{\circ}$, then the angle between the tangents at the ends of the radii is :

  1. $90^{\circ}$
  2. $40^{\circ}$
  3. $70^{\circ}$
  4. $60^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since tangents and radii are perpendicular at the point of contact, in the quadrilateral formed by the two radii and the tangents at their ends, we have two right angles at the two points of contacts.

Let the angle between the tangents be $x^o$. Then
$140 + 90 + 90 + x = 360 \Rightarrow x = 40^o$.
So option B 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

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

  1. $\dfrac{3\sqrt{3}}{2}$ cm
  2. $2\sqrt{3}$ cm
  3. $3\sqrt{3}$ cm

  4. 6 cm
Reveal answer Fill a bubble to check yourself
C 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 C 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

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values.

  1. Zero

  2. One

  3. Two

  4. Three
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Circumference = 10 units

m+n=10
n=10-m
't' is the length of the tangent.
$t^{2}=mn$
$t=\sqrt{m(10-m)}$
At $m=1, t=3$
At $m=2, t=4$
$\therefore$ Two values are possible for t.

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

A tangent from $P$, a point in the exterior of a circle touches circle at $Q$. If $OP=13$, $PQ=5$, then the diameter of the circle is ______________

  1. $576$
  2. $15$
  3. $8$
  4. $24$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Since tangent is perpendicular to the radius through the point of contact
so, $PQ \bot OQ$

therefore
${\left( {PQ} \right)^2} + {\left( {OQ} \right)^2} = {\left( {OP} \right)^2}$

$ = {\left( 5 \right)^2} + {r^2} = {\left( {13} \right)^2}$

$ = {r^2} = 169 - 25$

$\Rightarrow {r^2} = 144$

$\Rightarrow r = 12cm$

so, diameter of the circle $2 \times r$
$=2 \times 12$ $=24cm$

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

If $OA$ and $OB$ are the tangents to the circle ${x}^{2}+{y}^{2}-6x-8y+21=0$ drawn from the origin $O$, then $AB$ equals 

  1. ${ \dfrac { 17 }{ 3 } } $
  2. $\dfrac { 4 }{ 5 } \sqrt { 21 }$
  3. $11$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
equation of circle $\Rightarrow x^2+y^2-6x-8y+21=0$
radius $\Delta =\sqrt {9+16-21}=2$
$AB$ is chord of contact & its equation is
$x. x _1 +yy _1+9(x+x _1)+f(y+y _1)+c=0$
$(x _1, y _1)=(0,0)$
$0+0-3(x+0)-4(y+0)+21=0$
$3x+4y-21=0$
Perpendicular distance from $(3, 4)$ to line $l _1$
$CM=\dfrac {3(3)+4(4)-21}{\sqrt {9+16}}=\dfrac {4}{5}$
$AM=\sqrt {AC^2-CH^2}=\sqrt {4-\dfrac {16}{25}}=\dfrac {2}{5}\sqrt {21}$
$AB=2AM=\dfrac {4}{5}\sqrt {21} $ 
option $B$ is correct.


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

If 't$ _{1}$','t$ _{2}$','t$ _{3}$'are the lengths of the tangents drawnfrom centre of ex-circle to the circum circle of the $ \Delta A B C $, then- $ \frac { 1 } { t _ { 1 } ^ { 2 } } + \frac { 1 } { t _ { 2 } ^ { 2 } } + \frac { 1 } { t _ { 3 } ^ { 2 } } = $

  1. $ \frac { a b c } { a + b + c } $
  2. $ \frac { a b c } { a - b + c } $
  3. $ \frac { 2 a b c } { a + b + c } $

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

This is a known identity in triangle geometry relating the lengths of tangents from the excenter to the circumcircle.

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

If the distance from the origin of the centers of the three circles ${ x }^{ 2 }+{ y }^{ 2 }+2{ a } _{ i }x={ a }^{ 2 }\left( i=1,2,3 \right) $ are in G.P., then the length of the tangent drawn to them from any point on the circle ${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$ are in

  1. A.P.

  2. G.P.

  3. H.P.

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

The centers of the three given circles are $\left( -{ \alpha  } _{ 1 },0 \right) ,\left( -{ \alpha  } _{ 2 },0 \right) $ and $\left( -{ \alpha  } _{ 3 },0 \right) $.

the distance of the three points from the origin are ${ \alpha  } _{ 1 },{ \alpha  } _{ 2 }$ and ${ \alpha  } _{ 3 }$.
Given: ${ \alpha  } _{ 1 },{ \alpha  } _{ 2 }$ and ${ \alpha  } _{ 3 }$ are in G.P.
$\Rightarrow { { \alpha  } _{ 2 } }^{ 2 }={ \alpha  } _{ 1 }{ \alpha  } _{ 2 }$
Now, coordinate of any point on the circle ${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$ are $\left( a\cos { \theta  } ,a\sin { \theta  }  \right) $.
$\therefore$ The lengths of the tangents drawn from the point $\left( a\cos { \theta  } ,a\sin { \theta  }  \right) $ to the three given circles are
$\sqrt { 2{ \alpha  } _{ 1 }a\cos { \theta  }  } ,\sqrt { 2{ \alpha  } _{ 2 }a\cos { \theta  }  } $ and $\sqrt { 2{ \alpha  } _{ 3 }a\cos { \theta  }  } $
using (1) are in G.P.

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

If the length of the tangent drawn from any point on the circle $\displaystyle x^{2}+y^{2}+15x-17y+c^{2}=0$ to the circle $\displaystyle x^{2}+y^{2}+15x-17y+21=0 \ is \ \sqrt{5}$ units , then $c$ is equal to

  1. $-3$
  2. $3$
  3. $-4$
  4. $4$
Reveal answer Fill a bubble to check yourself
C,D Correct answer
Explanation
Required length
$\sqrt { { x }^{ 2 }+{ y }^{ 2 }+15x-17y+21-\left( { x }^{ 2 }+{ y }^{ 2 }+15x-17y+{ c }^{ 2 } \right)  } =\sqrt { 5 } $
$\Rightarrow \sqrt { 21-{ c }^{ 2 } } =\sqrt { 5 } \Rightarrow { c }^{ 2 }=16\Rightarrow c=\pm 4$
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 quadrilateral formed by the tangent from the point $(4, 5)$ to the circle $\displaystyle x^{2}+y^{2}-4x-2y-c=0$ with a pair of radii joining the points of contacts of these tangents is $8$ sq. units. The value of $c$ is

  1. $12$
  2. $-1$
  3. $3$
  4. $11$
Reveal answer Fill a bubble to check yourself
B,D Correct answer
Explanation

Given equation of circle is $x^2+y^2-4x-2y-c=0$


$(-g,-f)=(2,1)$

Radius $=\sqrt{g^2+f^2-c}$, $h$ of sub tangent 

Length of subtangent from point $(x _1,y _1) =\sqrt{x _1^2+y _1^2-4x _1-2y _1-c}$

 Area of quadrilateral = length of subtangent x radius

$\Rightarrow \sqrt { { 4 }^{ 2 }+{ 5 }^{ 2 }-4\times 4-2\times 5-c } \times \sqrt { 4+1+c } $

$ \Rightarrow { 8 }^{ 2 }=\left( 15-c \right) \left( 5+c \right) $

$\Rightarrow { c }^{ 2 }-10c-11=0$

$\Rightarrow c=11,-1$
Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

If $t _{i}$ is the length of the tangent to the circle $ x^{2}+ y^{2} + 2g _{i} x + 5 =0; i =1,2,3$ from any point and $g _{1}, g _{2}$ and $g _{3} $ are in A.P. and $A _{i} = (g _{i},- t _{i}^{2})$, then

  1. $A _{1}, A _{2}, A _{3} $are collinear
  2. $A _{2}$ is the mid-point of $A _{1}$ and $A _{3} $
  3. $ A _{1} A _{2} $ is perpendicular. to $A _{2} A _{3}$
  4. $A _{2}$ divides $A _{1} A _{3}$ in the ratio $2: 5$
Reveal answer Fill a bubble to check yourself
A,B Correct answer
Explanation

$t _{i}^{2} = x^{2} + y^{2} + 2g _{i}x + 5$  where $(x, y) $ is any point. 


Since  $g _{1}, g _{2}, g _{3}$ are in $A.P.$

$\Rightarrow 2g _{2} = g _{1} + g _{3}$
$\Rightarrow  2t _{2}^{2}  = t _{1}^{2} + t _{3}^{2} \Rightarrow  t _{1}^{2},t _{2}^{2} ,t _{3}^{2} $ are in $A.P.$
and $A _{2}$  is the mid-point of $A _{1}$ and $A _{3}$.

$\Rightarrow  A _{1}, A _{2}, A _{3}$  are collinear.

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

If the area of the quadrilateral formed by the tangent from the origin to the circle $x^{2} +y^{2} +6x -10y

+ c = 0$ and the pair of radii at the points of contact of these tangents to tbe circle is $8$ square units. then $c$ is a root of the equation

  1. $ c^{2} -32c + 64 = 0$.
  2. $ c^{2} -34c + 64= 0$.
  3. $c^{2}+ 2c -64 = 0 $.
  4. $ c^{2} + 34c -64 = 0$.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $OA, OB$ be the tangents from the origin to the given circle with centre $C(-3, 5)$
and radius .$\sqrt{9 + 25 -c}= \sqrt{ 34 -c} $
Then area of the quadribiteral $ OACB = 2 \times $ area of  $\triangle OAC = 2 \times (\dfrac 12) \times OA\times AC $
Now $OA =$ length of the tangent from the origin to the given circle $ = .\sqrt{C}$
and $AC =$ radius of the circle $=\sqrt{ 34 -c} $ so that. $\sqrt{C} \sqrt{34 -c} =8 $       ...(given)
$\Rightarrow  c (34 -c) =64 \Rightarrow c^{2} -34c+64=0$