Tag: tangents and intersecting chords

Questions Related to tangents and intersecting chords

The tangents drawn from origin to the circle ${ x }^{ 2 }+{ y }^{ 2 }-2ax-2by+{ b }^{ 2 }=0$ are perpendicular to each other, if

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

  1. $a-b=1$

  2. $a+b=1$

  3. ${ a }^{ 2 }-{ b }^{ 2 }=0$

  4. ${ a }^{ 2 }+{ b }^{ 2 }=0$

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Correct Option: C
Explanation:

Given circle equation ${x}^{2}+{y}^{2}-2ax-2by+{b}^{2}=0$
Center $: (a,b)$ and Radius $= \sqrt { {a}^{2}+{b}^{2}-{b}^{2} } $
Both tangents are drawn from origin and perpendicular to each other. So, two tangent are $x$ and $y$ axis.
Hence, $\sqrt { {a}^{2}+{b}^{2}-{b}^{2} } = a = b$
$\Rightarrow {a}^{2}={b}^{2}$
$\Rightarrow {a}^{2}-{b}^{2} = 0$ 

State whether the statement is true/false 

Two tangents $TP$ and $TQ$ are drawn to a circle with center $O$ from an external point $T$, then  $\angle PTQ=\angle OPQ$.

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

  1. True

  2. False

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Correct Option: B

Opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center 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

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Correct Option: A

If from a point P, two perpendicular tangents are drawn to the circle ${x^2} + {y^2} - 2x + 2y = 0$, then the coordinates of point P cannot be 

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

  1. $(3, - 1)$

  2. $(1,1)$

  3. $(\sqrt 3 + 1,0)$

  4. $(2,\sqrt 3 + 1)$

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Correct Option: D

Let $C _1$ and $C _2$ be two non concentric circles with $C _2$ lying inside $C _1$. A circle C lying inside $C _1$ touches $C _1$ internally and $C _2$ externally. The locus of the centre of the circle C is :

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

  1. Ellipse

  2. Circle

  3. Parabola

  4. None of these

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Correct Option: A

Let $C$ be the circle described $(x+a)^{2}+y^{2}=r^{2}$ where $0<r<a$ Let $m$ be the slope of the line through the origin that is tangent to $C$ at a point in the first quadrant. Then 

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

  1. $m=\dfrac{r}{\sqrt{a^{2}-r^{2}}}$

  2. $m=\dfrac{\sqrt{a^{2}-r^{2}}}{r}$

  3. $m=\dfrac{r}{a}$

  4. $m=\dfrac{a}{r}$

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Correct Option: A

Lines are drawn from the point $P(-1,3)$ to the circle $x^{2}+y^{2}-2x+4y-8=0$, which meets the circle at two points A and B. The minimum value of $PA+PB$ is

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

  1. $4$

  2. $6$

  3. $8$

  4. $16$

Show answer
Correct Option: C
Explanation:

$PA+PB \geq 2\sqrt {PA.PB}$   ...{A.M. $\geq $ G.M.}
$PA+PB \geq 2PT$   ...by tangent-secant theorem
$PA+PB \geq 2\sqrt{1+9-(-2)+12-8}=8$

A curve is such that the midpoint of the mid-point of the tangent intercepted between the point where the tangent is drawn and the point where the tangent is drawn and the point where the tangent meets y-axis, lies on the line $y=x$. If the curve passes through $(1,0)$, then the curve is

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

  1. $2y=x^2-x$

  2. $y=x^2-x$

  3. $y=x-x^2$

  4. $y=2(x-x^2)$

Show answer
Correct Option: C
Explanation:
Let $P\left(x,y\right)$ be a point on the curve then equation of the tangent is $Y-y=\dfrac{dy}{dx}\left(X-x\right)$

Given that tangent is drawn and the point where the tangent is drawn and the point where the tangent meets $y$-axis
$\Rightarrow\,x-$coordinate$=0$

$\Rightarrow\,X=0$

$\Rightarrow\, Y-y=\dfrac{dy}{dx}\left(X-x\right)$ becomes

$\Rightarrow\, Y-y=\dfrac{dy}{dx}\left(0-x\right)$ 

$\Rightarrow\, Y=y-x\dfrac{dy}{dx}$

$\therefore\,A=\left(0,y-x\dfrac{dy}{dx}\right)$

Given that  midpoint of the mid-point of the tangent intercepted between the point where the tangent is drawn and the point where the tangent is drawn and the point where the tangent meets $y$-axis, lies on the line $y=x$

$\therefore\,$Midpoint of the line $AP$ lies on the line $y=x$

Midpoint of the line $AP=\left(\dfrac{x+0}{2},\,\dfrac{y+y-x\dfrac{dy}{dx}}{2}\right)$ lies on  the line $y=x$

$\therefore\,x-$coordinate$=y-$coordinate

$\Rightarrow\,\dfrac{x+0}{2}=\dfrac{2y-x\dfrac{dy}{dx}}{2}$

$\Rightarrow\,x=2y-x\dfrac{dy}{dx}$ is a linear differential equation.
$\dfrac{dy}{dx}-\dfrac{2}{x}y=-1$

Integrating factor is $={e}^{\int{p\,dx}}={e}^{\int{\dfrac{-2}{x}\,dx}}={e}^{-\ln{x}}=\dfrac{1}{{x}^{2}}$

Now, $\dfrac{1}{{x}^{2}}\times\dfrac{dy}{dx}-\dfrac{1}{{x}^{2}}\times \dfrac{2}{x}y=\dfrac{-1}{{x}^{2}}$

$\Rightarrow\,\dfrac{1}{{x}^{2}}\dfrac{dy}{dx}-\dfrac{2}{{x}^{3}}y=\dfrac{-1}{{x}^{2}}$ 

$\Rightarrow\,\dfrac{1}{{x}^{2}}dy-\dfrac{2}{{x}^{3}}ydx=\dfrac{-dx}{{x}^{2}}$ 

Integrating both sides,we get

$\Rightarrow\,d\left(\dfrac{y}{{x}^{2}}\right)=d\left(\dfrac{1}{x}\right)$

$\Rightarrow\,d\left(\dfrac{y}{{x}^{2}}\right)=d\left(\dfrac{1}{x}\right)$

$\Rightarrow\,\dfrac{y}{{x}^{2}}=\dfrac{1}{x}+c$ is the required curve.

The curve passes through $\left(1,0\right)$

$\Rightarrow\,0=1+c$

$\Rightarrow\,c=-1$

$\Rightarrow\,\dfrac{y}{{x}^{2}}=\dfrac{1}{x}-1$

$\Rightarrow\,\dfrac{y}{{x}^{2}}=\dfrac{1-x}{x}$

$\Rightarrow\,y=x-{x}^{2}$ is the required equation of the curve passing through $\left(1,0\right)$

The locus of the centre of a circle touching the lines $x+2y=0$ and $x-2y=0$ is

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

  1. $xy=0$

  2. $x=0$

  3. $y=0$

  4. none of these

Show answer
Correct Option: A
Explanation:

let $(h,k)$ be the center
Then distance from center to both lines will be equal
$\left| \dfrac { h+2k }{ \sqrt { 5 }  }  \right| =\left| \dfrac { h-2k }{ \sqrt { 5 }  }  \right| $
$\Rightarrow hk=0$

Ans: A

Consider a circle, $x^{2}+y^{2}=1$ and point $P\left(1,\sqrt{3}\right).PAB$ is secant drawn from $P$ intersecting circle in $A$ and $B$ (distinct) then range of $\left|PA\right|+\left|PB\right|$is 

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

  1. $\left[2\sqrt{3},4\right]$

  2. $\left(2\sqrt{3},4\right]$

  3. $\left(0,4\right]$

  4. $\left(0,2\sqrt{3}\right)$

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Correct Option: A