Chord of Contact

Q. The equation of the chord of contact of the tangents drawn from the point (3, 4) to the parabola $y^2=2x$ is .

1. x-4y+3=0
2. x-4y-3=0
3. x+4y+3=0
4. x+4y-3=0

Q. The locus of mid points of the chords of the parabola $y^2=4(x+1)$ which are parallel to $3x=4y$ is

1. $5y-8=0$
2. $3x+10=0$
3. $5y+9=0$
4. $3y-8=0$

Q. The mid point of line joining the common points of the line $2x-3y+8=0$ and $y^2=8x,$ is

1. $(3,2)$
2. $(5,6)$
3. $(4,-1)$
4. $(2,-3)$

Q. A tangent is drawn at any point $(x_1,y_1)$ other than the vertex on the parabola $y^2=4ax$. Tangents are drawn from any point of this tangent to the circle $x^2+y^2=a^2$ such that all the chords of contact passes 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. If the tangent at the point $P(2,4)$ to the parabola $y^2=8x$ meets the parabola $y^2=8x+5$ at $Q$ and $R$, then the sum of the coordinates of the mid point of $QR$ is ​​

Q. Let from any point $P$ on the line $y=x,$ two tangents are drawn to the circle $(x-2{)}^2+y^2=1.$ Then the chord of contact of $P$ with respect to given circle always passes through a fixed point, whose coordinates are given by

1. $(\frac{3}{2},\frac{1}{4})$
2. $(-\frac{3}{2},\frac{1}{4})$
3. $(-\frac{3}{2},\frac{1}{2})$
4. $(\frac{3}{2},\frac{1}{2})$

Q. Points $A$ and $B$ lie on the parabola $y=2x^2+4x-2$, such that origin is the mid-point of the segment $AB$. If $l$ is the length of the line segment $AB$, then the value of $l^2$ is

Q. The chords of contact of the pair of tangents drawn from each point on the line
$2x+y=4$ to the circle $x^2+y^2=1$ pass through the point ___.

1. (2, 1)
2. (1, 2)
3. $(\frac{1}{2},\frac{1}{4})$
4. $(\frac{1}{4},\frac{1}{2})$

Q. Let x(x - a) + y(y - 1) be a circle. If two chords from (a, 1) bisected by the x-axis are drawn to the circle then the condition required is

1. $a^2$ = 8
2. $a^2$ $<$ 8
3. $a^2$ $>$ 8
4. a=0

Q. The tangents to the curve $y=(x-2{)}^2-1$ at its points of intersection with the line $x-y=3$, intersect at the point :

1. $(\frac{5}{2},-1)$
2. $(-\frac{5}{2},-1)$
3. $(\frac{5}{2},1)$
4. $(-\frac{5}{2},1)$

Q. Tangents are drawn from the point $P(2,2)$ to the circle $x^2+y^2=1,$ touching the circle at $A$ and $B.$ Then equation of circumcircle of $△PAB$ is

1. $x^2+y^2+2x+2y=0$
2. $x^2+y^2-2x-2y=0$
3. $x^2+y^2+2x-2y=0$
4. $x^2+y^2-2x+2y=0$

Q. Tangents are drawn from different points on the line $x-y+10=0$ to the parabola $y^2=4x$. If the chords of contact pass through a fixed point, then the coordinates of the fixed point is

1. $(2,5)$
2. $(5,2)$
3. $(10,2)$
4. $(2,10)$

Q. Tangents are drawn to $x^2+y^2=1$ from any arbitrary point $P$ on the line $2x+y-4=0$. The corresponding chord of contact passes through a fixed point whose coordinates are

1. $(\frac{1}{4},\frac{1}{2})$
2. $(\frac{1}{2},1)$
3. $(\frac{1}{2},\frac{1}{4})$
4. $(1,\frac{1}{2})$

Q. If a tangent is drawn to the parabola $y^2=4x$ through the point $(-2,1)$ then the points of contact is/are

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

Q. If a triangle is formed by any three tangents of the parabola $y^2=4ax$ whose two of its vertices lie on $x^2=4by$, then third vertex lie on

1. $(x-1{)}^2=4ay$
2. $x^2=16ay$
3. $x^2=4by$
4. $(x+1{)}^2=4ay$

Q. If the tangents are drawn to the circle $x^2+y^2=12$ at the point where it meets the circle $x^2+y^2-5x+3y-2=0$, then the point of intersection of these tangents is

1. $(6,-\frac{18}{5})$
2. $(-6,\frac{18}{5})$
3. $(7,-\frac{18}{6})$
4. $(8,-\frac{18}{5})$

Q. If $△PAB$ is right angle at $P$, where $A(2,3)$ and $B(5,7)$, then which of the following is/are correct?

1. Equation of circumcircle of $△PAB$ is $x^2+y^2-7x-10y+31=0$
2. Maximum possible area of $△PAB$ is $\frac{25}{4}\text{sq. units}.$
3. When area of $△PAB$ is $6\text{sq. units}$, then there are $4$ possible values of $P.$
4. When area of $△PAB$ is $6\text{sq. units}$, then there are $2$ possible values of $P.$

Q. The coordinates of the point of intersection of tangents drawn to $y^2=4ax$ at the point of intersection with the line $x\cos \alpha +y\sin \alpha -p=0$ is

1. $(p\tan \alpha ,2a\sec \alpha )$
2. $(-p\sec \alpha ,-2a\tan \alpha )$
3. $(2a\sec \alpha ,p\tan \alpha )$
4. $(p\sec \alpha ,2a\tan \alpha )$

Q. The locus of the point, whose chord of contact w.r.t the circle $x^2+y^2=a^2$ makes an angle $2\alpha$ at the centre of the circle is

1. $x^2+y^2=2a^2$
2. $x^2+y^2=a^{2}co{s}^2\alpha$
3. $x^2+y^2=a^{2}se{c}^2\alpha$
4. $x^2+y^2=a^{2}ta{n}^2\alpha$

Q. Tangents are drawn from any point on the line $x+4a=0$ to the parabola $y^2=4ax$. Then the angle subtended by the chord of contact at the vertex will be .

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{6}$

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

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

Q. Let $P$ be any moving point on the circle $S_1:x^2+y^2-2x-1=0.$ A chord of contact is drawn from the point $P$ to the circle $S:x^2+y^2-2x=0.$ If $C$ is the centre and $A,B$ are the points of contact of circle $S,$ then the locus of the circumcentre of $△CAB$ is

Q. If $L_1$ is the equation of the chord of the circle $x^2+y^2-6x-4y+4=0$ passing through $(3,4)$ and farthest from the centre, then which of the following is/are correct?

1. Equation of $L_1:2x-y=2$
2. Length of the chord is $2\sqrt{5}$
3. Equation of $L_1:y=4$
4. Length of the chord is $\sqrt{5}$

Q.

Find the equation of the chord of contact of tangents to the parabola
$y^2=4x$ from the point $P(3,4)$.

1. $x-2y-6=0$

2. $x+2y+6=0$

3. $x+y+3=0$

4. $x-2y-3=0$

Q. Let $P$ be any moving point on the circle $S_1:x^2+y^2-2x-1=0.$ A chord of contact is drawn from the point $P$ to the circle $S:x^2+y^2-2x=0.$ If $C$ is the centre and $A,B$ are the points of contact of circle $S,$ then the locus of the circumcentre of $△CAB$ is

1. $(x-1{)}^2+y^2=\frac{1}{2}$
2. $x^2+(y-1{)}^2=\frac{1}{2}$
3. $(x-1{)}^2+y^2=1$
4. $x^2+(y-1{)}^2=\frac{1}{\sqrt{2}}$

Q. Locus of mid points of normal chords of the parabola $y^2=4ax$ is :

1. $8a^4+4a^2{y}^2+y^2(y^2-4ax)=0$
2. $12a^4+7a^2{y}^2+y^2(y^2-4ax)=0$
3. $10a^4+6a^2{y}^2+y^2(y^2-8ax)=0$
4. $8a^4+9a^2{y}^2+y^2(y^2-7ax)=0$

Q. The line $x=y$ touches a circle at the point $(1,1).$ If the circle also passes through the point $(1,-3),$ then its radius (in units) is :

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