Tag: angle made by a chord and a tangent

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Questions Related to angle made by a chord and a tangent

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

Consider a circle $x^2+y^2=3$. Secants are drawn from (-2,0) to the circle which make an intercept of $2\sqrt{2}$ units on the circle. Identify the correct statements ?

  1. The combined equation of the secants is $x^2-4y^2+2x+1=0$
  2. The combined equation of the secants is $x^2-4y^2+x+1=0$
  3. Angle between the secants is $60^{o}$
  4. Angle between the secants is $30^{o}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the condition for secants from a point to a circle and the given intercept length, the combined equation of the secants can be derived as x^2 - 4y^2 + 2x + 1 = 0.

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).

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

Let  $ABCD$  be a quadrilateral in which $A B | C D , A B \perp A D \text { and } A B = 3 C D$. The area of quadrilateral  $ABCD$  is  $4.$  The radius of a Circle touching all the sides of quadrilateral is = ?

  1. $\sin \frac { \pi } { 12 }$
  2. $\sin \frac { \pi } { 6 }$
  3. $\sin \frac { \pi } { 4 }$
  4. $\sin \frac { \pi } { 3 }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given AB || CD, AB perpendicular to AD, and AB = 3CD, this is a right trapezoid. With area 4, we find the height and side lengths. A circle touches all sides if the sum of opposite sides is equal, which leads to the radius calculation via the geometry of the trapezoid.