Chord of Contact

Q. Let $B$ be the centre of the circle $x^2+y^2-2x+4y+1=0$. Let the tangents at two points $P$ and $Q$ on the circle intersect at the point $A(3,1).$ Then $8\cdot \left(\frac{\text{area}△APQ}{\text{area}△BPQ}\right)$ is equal to

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. Let $PQ$ be a diameter of the circle $x^2+y^2=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line, $x+y=2$ respectively, then the maximum value of $\alpha \beta$ is

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 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. Number of common tangents of $y=x^2$ and $y=–x^2+4x-4$ is

1. 1
2. 2
3. 3
4. 4

Q. If the chord of contact of the tangents from a point $P$ to the parabola $y^2=4ax$ touches the parabola $x^2=4by$, then the locus of $P$ is

1. $xy=ab$
2. $xy=2ab$
3. $xy=-ab$
4. $xy=-2ab$

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. $TP$ and $TQ$ are tangents to the parabola $y^2=4ax$ at $P$ and $Q.$ If the chord $PQ$ passes through the fixed point $(-a,b),$ then the locus of $T$ is

1. $ay=2b(x-b)$
2. $by=2a(x-a)$
3. $ax=2b(y-b)$
4. $bx=2a(y-a)$

Q. A variable chord $PQ$ of the parabola $y^2=4ax$ is drawn parallel to line $y=x$. Then the locus of point of intersection of normals at $P$ and $Q$ is:

1. $2x-y-12a=0$
2. $2x+y-a=0$
3. $x-2y-4a=0$
4. $3x-y-8a=0$

Q. Points $A$ and $B$ lies 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.

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. Tangent are drawn from the points on the line $x-y-5=0$ to $x^2+4y^2=4$, then all the chords of contact pass through a fixed point, whose coordinate are

1. $(\frac{4}{5},-\frac{1}{5})$
2. $(\frac{4}{5},\frac{1}{5})$
3. $(-\frac{4}{5},\frac{1}{5})$
4. None of the these

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. 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.

The number of common tangents of the circles ${\left(x+2\right)}^2+{\left(y-2\right)}^2=49$ and ${\left(x-2\right)}^2+{\left(y+1\right)}^2=4$ are

1. $0$

2. $1$

3. $3$

4. $4$

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. If a circle cuts the rectangular hyperbola xy = 1 in the points $(x_r,y_r)$; $r=1,2,3,4$, then $x_1{x}_2{x}_3{x}_4=y_1{y}_2{y}_3{y}_4$is equal to

1. 2
2. 1
3. 3
4. 4

Q. The angle between pair of tangents drawn from any point on the circle $x^2+y^2=a^2$ upon the circle $x^2+y^2=b^2$ is $\frac{\pi }{3}$. Then, the locus of mid point of chord of contact is

1. $x^2+y^2=\frac{5b^2-a^2}{4}$
2. $x^2+y^2=\frac{3a^2-b^2}{4}$
3. $x^2+y^2=\frac{4a^2-b^2}{4}$
4. $x^2+y^2=\frac{5a^2-b^2}{4}$

Q. A circle of radius $\sqrt{5}$ units has diameter along the angle bisector of the lines $x+y=2$ and $x-y=2$. If chord of contact from the origin makes an angle of ${45}^{\circ }$ with the positive direction of $x$-axis, then the equation of the circle is

1. $(x+2{)}^2+(y-2{)}^2=5$
2. $(x-2{)}^2+y^2=5$
3. $(x-2{)}^2+(y+2{)}^2=5$
4. $(x+2{)}^2+y^2=5$

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 curve c has a property that if a tangent at any point, say P, meets the axes at A and B then P is the midpoint of AB if the curve passes through (1, 1) find its equation.

Q. Find the area enclosed between the parabola $4y$ =$3x^2$ and the straight line $3x-2y+12=0$.

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

Q. The value of $x$ which satisfy the equation $(x-1{)}^{{\log }_3{x}^2-2{\log }_{x}9}=(x-1{)}^7$ is equal to

1. $3,17$
2. $\frac{1}{\sqrt{3}},81$
3. $13,21$
4. $noneofthese$

Q.

Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

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

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

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

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