Tag: touching circles

Quizzes (0) Questions (86)

Questions Related to touching circles

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

The equation to the locus of the point of intersection of any two perpendicular tangents to $x^{2}+ y^{2} = 4$ is

  1. $\mathrm{x}^{2}+\mathrm{y}^{2}=8$
  2. $\mathrm{x}^{2}+\mathrm{y}^{2}=12$
  3. $\mathrm{x}^{2}+\mathrm{y}^{2}=16$
  4. $\mathrm{x}^{2}+\mathrm{y}^{2}=4\sqrt{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of the tangent to the circle $x^2+y^2=4$ is

$y=mx+2\sqrt{1+m^2}$
$P(h,k)$ lies on the tangent, then
$k-mh=2\sqrt{1+m^2}$
or, $(k-mh)^2=4(1+m^2)$
or, $m^2(h^2-4)-2mhk+k^2-4=0$
This is the quadratic equation in $m.$ Let $m _1$ and $m _2$ be roots
$m _1m _2=\cfrac{k^2-4}{h^2-4}=-1$
or, $k^2-4=-h^2+4$
or, $h^2+k^2=8$
Therefore, Equation to the locus of the intersection of any two perpendicular tangents is
$x^2+y^2=8$
Hence, A is the correct option.

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

Two secants PAB and PCD are drawn to a circle from an outside point P. Then, which of the following is true?

  1. PA. PB =PC +CD

  2. PA. PB =PC. PD

  3. PA+PB=PC+PD

  4. PA-PB = PC. CD
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

By the power of a point theorem, for two secants PAB and PCD drawn from an external point P to a circle, the product of the segments of the secants is equal: PA * PB = PC * PD.

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.