Length of Chord of Contact

Q. The length of common chord of two intersecting circles is $30$cm. If the diameters of these two circles be $50$ cm and $34$ cm, then the distance between their centres is

Q.

The radius of a circle is 30 cm. Find the length of an arc of this circle, if the length of the chord of the arc is 30 cm.

Q. Tangents $PA$ and $PB$ are drawn to $x^2+y^2=4$ from the point $P(3,0).$ Then the area (in sq. units) of $△PAB$ is

1. $\frac{10}{9}\sqrt{5}$
2. $\frac{5}{9}\sqrt{5}$
3. $\frac{20}{3}\sqrt{5}$
4. $\frac{1}{3}\sqrt{5}$

Q.

Tangents are drawn from (4, 4) to the circle $x^2+y^2-2x-2y-7=0$ to meet the circle at A and B. The length of the chord AB is

1. $2\sqrt{3}$

2. $3\sqrt{2}$

3. $2\sqrt{6}$

4. $6\sqrt{2}$

Q. A tangent is drawn at any point $(x_1,y_1)$ other than vertex on the parabola $y^2=4ax$. If tangents are drawn from any point on this tangent to the circle $x^2+y^2=a^2$ such that all the chords of contact pass through a fixed point $(x_2,y_2)$, then

1. $x_1,a,x_2$ are in G.P
2. $\frac{y_1}{2},a,y_2$ are in G.P.
3. $-4,\frac{y_1}{y_2},\frac{x_1}{x_2}$ are in G.P.
4. $x_1{x}_2+y_1{y}_2=a^2$

Q. For the line 3x + 2y = 12 and the circle $x^2+y^2-4x-6y+3=0,$ which of the following statements is true

1. Line is a tangent to the circle
2. Line is a chord of the circle
3. Line is a diameter of the circle
4. None of these

Q. The sphere $|\overrightarrow{r}|=5$ is cut by the plane $\overrightarrow{r}\cdot (\hat{i}+\hat{j}+\hat{k})=3\sqrt{3}$. The radius of the circular section formed is

Q.

One extremity of a focal chord of the parabola $y^2=16x$ is A(1, 4). Then the length of the focal chord at A is

1. 25/4

2. 25/2

3. 15/2

4. 25

Q. A circle is circumscribed about a triangle $ABC$. A tangent is drawn to the circle through the point $B$ to intersect the extension of the side $CA$ beyond the point $A$ at the point $D$. Let $P$ be the perimeter of triangle $ABC$. Given $AB+AD=AC,CD=3$ and $\angle BAC={60}^{\circ }$ then the value of $(3-\sqrt{2})P$ is

1. 3
2. 6
3. 9
4. None of these

Q. Let the tangents drawn from the origin to the circle, $x^2+y^2-8x-4y+16=0$ touch it at the points $A$ and $B$. The $(AB{)}^2$ is equal to :

1. $\frac{32}{5}$
2. $\frac{64}{5}$
3. $\frac{52}{5}$
4. $\frac{56}{5}$

Q. A pair of tangents are drawn from origin to the circle $x^2+y^2-8x+12=0$. Another tangent is drawn to this circle such that this circle will become the incircle of the triangle which is formed by these three tangents. What will be the area of the triangle formed?

1. $12$
2. $12\sqrt{3}$
3. $16\sqrt{3}$
4. $16$

Q. A pair of tangents $OA,OB$ $(O$ is origin$)$ is drawn to a circle whose centre is $C$ and radius is $3$ units. If the combined equation of $OA$ and $OB$ is $2x^2-3xy+y^2=0,$ then the area (in sq. units) of the quadrilateral $OACB$ is equal to

1. $3(3+\sqrt{10})$
2. $9(3+\sqrt{10})$
3. $6(3+\sqrt{10})$
4. $12(3+\sqrt{10})$

Q. The length of the chord of contact with respect to the point on the director circle of circle $x^2+y^2+2ax-2by+a^2-b^2=0$ is $k|b|$ units. Then the value of $k$ is

Q. The length of the chord $AB$ of the circle $x^2+y^2-6x+8y-13=0$ whose midpoint is $(2,-3)$ is (units)

Q. Length of chord of contact drawn from $(0,0)$ to the circle $x^2+y^2+16x+12y+8=0$ is $0.8\sqrt{k}$ units, then $k=$

Q.

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

(i) p: Each radius of a circle is a chord of the circle.

(ii) q: The centre of a circle bisects each chord of the circle.

(iii) r: Circle is a particular case of an ellipse.

(iv) s: If x and y are integers such that x > y, then –x < –y.

(v) t: is a rational number.

Q. Let tangents are drawn from $P(3,4)$ to the circle $x^2+y^2=a^2,$ touching the circle at $A$ and $B.$ If area of $△PAB$ is $\frac{192}{25}$ sq. units, then the absolute value of $a$ is

Q. Circle $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius $6$ in such a manner that their common chord is of maximum length and has slope equal to $\frac{1}{2}.$ Then the co-ordinates of the centre of the circle(s) $C_2$ is (are)

1. $(2,4)$
2. $(2,-4)$
3. $(-2,-4)$
4. $(-2,4)$

Q. If two chords drawn from the point $A(4,4)$ to the parabola $x^2=4y$ are bisected by line $y=mx$, the interval in which m lies is

1. None of these
2. $(-\infty ,-\sqrt{2})\cup (\sqrt{2},\infty )$
3. $(-\infty ,-2\sqrt{2}-2)\cup (2\sqrt{2}-2,\infty )$
4. $(-2\sqrt{2},2\sqrt{2})$

Q. The common tangent to the circles $x^2+y^2-6x=0$ and $x^2+y^2+2x=0$ forms

1. isosceles triangle
2. Right angle isosceles triangle
3. Equilateral triangle
4. Right angle triangle

Q. If one end of a diameter of the circle $x^2+y^2-4x-6y+11=0$ be (3, 4), then the other end is

1. (0, 0)
2. (1, 1)
3. (1, 2)
4. (2, 1)

Q.

For a question like A U B intersection C, if no brackets are specified, how do you approach. Is there a rule such as BODMAS for Operations on Sets also?

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(I)}& \text{The length of the common chord of}& & \\ & \text{two circle of radii}3\text{and}4\text{units which}& & \\ & \text{intersect orthogonally is}\frac{k}{5},\text{then k is}& \text{(P)}& 1\\ \text{(II)}& \text{The circumference of the circle}& & \\ & x^2+y^2+4x+12y+p=0\text{is bisected}& & \\ & \text{by the circle}{x}^2+y^2-2x+8y-q=0,& \text{(Q)}& 2\\ & \text{then}p+q\text{equals to}& & \\ \text{(III)}& \text{The number of distinct chords of}& & \\ & x^2+y^2+2x+4y=0\text{which passes}& \left(R\right)& 12\\ & \text{through the point}(2,1)\text{and bisected}& & \\ & \text{by}x\text{- axis is}& & \\ \text{(IV)}& \text{One of the diameter of the circle}& \left(S\right)& 24\\ & \text{circumscribing the rectangle}ABCD& & \\ & \text{is}4y=x+7.\text{If}A\text{and}B\text{are}(-3,4)& & \\ & \text{and}(5,4)\text{respectively, then the area}& & \\ & \text{of the rectangle}ABCD\text{is}& & \\ & & \left(T\right)& 32\\ & & \left(U\right)& 36\end{array}$

Which of the following option is INCORRECT ?

1. $\left(I\right)\to \left(R\right)$
2. $\left(II\right)\to \left(U\right)$
3. $\left(III\right)\to \left(P\right)$
4. $\left(IV\right)\to \left(T\right)$

Q.

RS is the chord of contact of the point P on a circle center at O. if $m_1=slopeof\overline{RS}and{m}_2=slopeof\overline{OP}$

1. $m_1+m_2=0$

2. $\frac{m_1}{m_2}=+1$

3. $m_1-m_2=1$

4. $m_1.m_2=-1$

Q. Let tangents are drawn from $P(3,4)$ to the circle $x^2+y^2=a^2,$ touching the circle at $A$ and $B.$ If area of $△PAB$ is $\frac{192}{25}$ sq. units, then the absolute value of $a$ is

Q. The tangent lines to the circle $x^2+y^2-6x+4y=12$ which are parallel to the line $4x+3y+5=0$ are given by

1. $4x+3y+5=0,4x+3y-25=0$
2. $4x+3y-31=0,4x+3y+19=0$
3. $4x+3y-17=0,4x+3y+13=0$
4. None of these

Q. A tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ at any point $P$ meets the line $x=0$ at a point $Q$. Let $R$ be the image of $Q$ in the line $y=x$. Then the circle, whose extremities of a diameter are $Q$ and $R$, passes through a fixed point whose coordinates are

1. $(5,0)$
2. $(0,0)$
3. $(3,0)$
4. $(4,0)$

Q. Let the tangents drawn from the origin to the circle, $x^2+y^2-8x-4y+16=0$ touch it at the points $A$ and $B$. The $(AB{)}^2$ is equal to :

1. $\frac{32}{5}$
2. $\frac{64}{5}$
3. $\frac{52}{5}$
4. $\frac{56}{5}$

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(I)}& \text{The length of the common chord of}& & \\ & \text{two circle of radii}3\text{and}4\text{units which}& & \\ & \text{intersect orthogonally is}\frac{k}{5},\text{then k is}& \text{(P)}& 1\\ \text{(II)}& \text{The circumference of the circle}& & \\ & x^2+y^2+4x+12y+p=0\text{is bisected}& & \\ & \text{by the circle}{x}^2+y^2-2x+8y-q=0,& \text{(Q)}& 2\\ & \text{then}p+q\text{equals to}& & \\ \text{(III)}& \text{The number of distinct chords of}& & \\ & x^2+y^2+2x+4y=0\text{which passes}& \left(R\right)& 12\\ & \text{through the point}(2,1)\text{and bisected}& & \\ & \text{by}x\text{- axis is}& & \\ \text{(IV)}& \text{One of the diameter of the circle}& \left(S\right)& 24\\ & \text{circumscribing the rectangle}ABCD& & \\ & \text{is}4y=x+7.\text{If}A\text{and}B\text{are}(-3,4)& & \\ & \text{and}(5,4)\text{respectively, then the area}& & \\ & \text{of the rectangle}ABCD\text{is}& & \\ & & \left(T\right)& 32\\ & & \left(U\right)& 36\end{array}$

Which of the following option is INCORRECT ?

1. $\left(I\right)\to \left(R\right)$
2. $\left(II\right)\to \left(U\right)$
3. $\left(III\right)\to \left(P\right)$
4. $\left(IV\right)\to \left(T\right)$

Q. The common tangent to the circles $x^2+y^2-6x=0$ and $x^2+y^2+2x=0$ forms

1. isosceles triangle
2. Right angle triangle
3. Right angle isosceles triangle
4. Equilateral triangle