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

For the two circles ${ x }^{ 2 }+{ y }^{ 2 }=16$ and ${ x }^{ 2 }+{ y }^{ 2 }-2y=0$ there is/are

  1. One pair of common tangents

  2. Only one common tangent

  3. Three common tangents

  4. No common tangent
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The centres and radii of given circles are ${ C } _{ 1 }\left( 0,0 \right) ,{ r } _{ 1 }=4$ and ${ C } _{ 2 }\left( 0,1 \right) ,{ r } _{ 2 }=\sqrt { 0+1 } =1$
Now, ${ C } _{ 1 }{ C } _{ 2 }=\sqrt { 0+{ \left( 0-1 \right)  }^{ 2 } } =1$
and ${ r } _{ 1 }-{ r } _{ 2 }=4-1=3$
$\therefore { C } _{ 1 }{ C } _{ 2 }<{ r } _{ 1 }-{ r } _{ 2 }$
Hence, second circle lies inside the first circle.

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 outside a circle, one tangent and one secant are drawn. The length of exterior part of secant is $7$ cm and that of interior part is $9$ cm. Find the length of tangent segment.

  1. $10.6$ cm
  2. $10.9$ cm
  3. $11.2$ cm
  4. $11.6$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let length of tangent be $l$

Length of exterior part of secant $=m=7 $ cm
Length of interior part of secant $=n=9 $ cm
Now using the secant intersection theorem, we have
${ l }^{ 2 }=m(m+n)\ \Rightarrow { l }^{ 2 }=7(7+9)\ \Rightarrow { l }^{ 2 }=112 $
$\Rightarrow l=\sqrt { 112 } =10.6$ cm
Option A is correct.

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

Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is $60^0$. Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of tangents.

  1. 4 cm

  2. 6 cm

  3. 8 cm

  4. 10 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Join O and P.
Now, in triangles OPQ and OPR,
OP = OP (Common)
OQ = OR (radius of circle)
PQ = PR (tangents from single point)

Hence OPQ and OPR are congruent triangles.
$\angle OPQ = \angle OPR = 30^{\circ}$


Thus in triangle OPQ, $\dfrac{OQ}{OP} = Sin 30$

OP = $\dfrac{4}{Sin30}$

OP = 8 cm

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 pair of tangents are drawn from a point $P$ to the circle $x^{2} + y^{2} = 1$. If the tangents make an intercept of $2$ on the line $x = 2$, the locus of $P$ is

  1. Straight line

  2. Pair of lines

  3. Circle

  4. Parabola
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let $P$ be $(h, k)$ then by $SS _{1} = T^{2}$ the equation of pair of tangents drawn from $P$ to $x^{2} + y^{2} = 1$ is
$(h^{2} + k^{2} - 1)(x^{2} + y^{2} - 1) = (hx + ky - 1)^{2}$
Its intersection with the line $x = 1$ is given by
$y^{2}(h^{2} + k^{2} - 1) = (ky + h - 1)^{2}$
or $y^{2}(h^{2} - 1) - 2yk (h - 1) - (h - 1)^{2} = 0 .....(1)$
It is a quadratic in $y$ and we are given that length of intercept is $1 \therefore y _{1} - y _{2} = 2$
or $(y _{1} + y _{2})^{2} - 4y _{1}y _{2} = 1$
or $\left [\dfrac {2k(h - 1)}{h^{2} - 1}\right ]^{2} + 4\dfrac {(h - 1)^{2}}{h^{2} - 1} = 4$
or $\dfrac {4k^{2}}{(h + 1)^{2}} + \dfrac {4(h - 1)}{h + 1} = 4$
or $k^{2} = (h + 1)^{2} - (h^{2} - 1) = 2h + 2$
Hence the locus of $(h, k)$ is $y^{2} =2(x + 1)$ which represents a parabola.

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

Tangents $TP$ and $TQ$ are drawn from a point $T$ to circle $x^{2}+y^{2}=a^{2}$. If the point $T$ lies on the line $px+qy=r$, then locus of the centre of circumcircle of $\triangle TPQ$ is

  1. straight line

  2. circle

  3. parabola

  4. ellipse
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The circumcircle of triangle TPQ has the segment OT as its diameter, where O is the center of the circle and T is the external point. If T lies on a line, the locus of the midpoint of OT will also be a straight line.

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

Tangents PA and PB are drawn to the cicle $S\, \equiv \,{x^2}\, + \,{y^2}\, - \,2y\, - \,3\, = \,0$ from the point $P(3, 4)$. Which of the following alternative(s) is/are correct ?

  1. The power of point $P(3, 4)$ with respect to circle $S=0$ is $14$.
  2. The angle between tangents from $P(3, 4)$ to the circle $S=0$ is $\frac{\pi }{3}$
  3. The equation of circumcircle of $\Delta PAB\,$ is ${x^2}\, + \,{y^2}\, - \,3x\, - \,5y\, + \,4\, = 0$
  4. The area of quadrilateral $PACB$ is $3\sqrt 7 $ square units where C is the centre of circle $S = 0$.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The power of a point (x1, y1) with respect to a circle x^2 + y^2 + 2gx + 2fy + c = 0 is x1^2 + y1^2 + 2gx1 + 2fy1 + c. For P(3,4) and S = x^2 + y^2 - 2y - 3 = 0, Power = 3^2 + 4^2 - 2(4) - 3 = 9 + 16 - 8 - 3 = 14.

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

$y=mx+b$ is a tangent to the circle ${x}^{2}+{y}^{2}-6x=16\ if\ \left (3\ m+b\right)^{2}=5\left (1+{m}^{2}\right)$.

  1. True

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

The condition for a line y = mx + b to be tangent to a circle x^2 + y^2 - 2gx - 2fy + c = 0 is (mg + f + b)^2 = r^2(1 + m^2). For x^2 + y^2 - 6x - 16 = 0, the center is (3, 0) and r^2 = 16 + 9 = 25. Substituting g=3, f=0, r^2=25 gives (3m + b)^2 = 25(1 + m^2), not 5(1+m^2).