Tag: tangents and intersecting chords

Questions Related to tangents and intersecting chords

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

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

  1. $1$

  2. $2$

  3. $4$

  4. $6$

Show answer
Correct Option: A
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 $2r$.
$\therefore d=\left | \dfrac{-10-16}{\sqrt{5^{2}+12^{2}}} \right |=\left | \dfrac{26}{13} \right |=2$
$\therefore \ d=2r=2$
$\Rightarrow r=radius=1$

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

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

  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. $\mathrm{x}^{2}+\mathrm{y}^{2}=4\sqrt{3}$

Show answer
Correct Option: A
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.

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

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

  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

Show answer
Correct Option: B
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 $k\left( { y }^{ 2 }{ -a }^{ 2 } \right) =2xy$

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

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

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

  2. $\displaystyle \frac{\pi}{6}$

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

  4. $\displaystyle \frac{\pi}{8}$

Show answer
Correct Option: C
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}$

The number of tangents that can be drawn from (1, 2) to $x^2+y^2=5$ is

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

  1. 1

  2. 2

  3. 3

  4. 0

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Correct Option: A

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

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

  1. PA. PB =PC +CD

  2. PA. PB =PC. PD

  3. PA+PB=PC+PD

  4. PA-PB = PC. CD

Show answer
Correct Option: B

State true or false
The angle between two tangents to circle may be ${0^0}$

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

true 

the angles between tangents to a circle can be zero if the tangents are parallel {if the tangents are drawn on opposite sides of a diameter}

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

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

  1. True

  2. False

Show answer
Correct Option: B
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. 

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

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

  1. True

  2. False

Show answer
Correct Option: A
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.

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.

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

  1. $\displaystyle { 30 }^{ \circ }$

  2. $\displaystyle { 60 }^{ \circ }$

  3. $\displaystyle { 90 }^{ \circ }$

  4. $\displaystyle { 150 }^{ \circ }$

Show answer
Correct Option: D