Equation of Tangent in Point Form

Q. Two tangents are drawn from the point $P(-1,1)$ to the circle $x^2+y^2-2x-6y+6=0.$ If these tangents touch the circle at points $A$ and $B$, and if $D$ is a point on the circle such that length of the segments $AB$ and $AD$ are equal, then the area of the triangle $ABD$ is equal to

1. $2$
2. $(3\sqrt{2}+2)$
3. $4$
4. $3(\sqrt{2}-1)$

Q. If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is:

1. $x^2+y^2-2xy=0$
2. $x^2+y^2-2x^2{y}^2=0$
3. $x^2+y^2-4x^2{y}^2=0$
4. $x^2+y^2-16x^2{y}^2=0$

Q.

ABCD is a square of side 1 unit. A circle passes through vertices A, B of the square and the remaining two vertices of the square lie out side the circle. The length of the tangent drawn to the circle from vertex D is 2 units. The radius of the circle is

1. 

2. 

3. 

4. 

Q. If the tangent to the circle $x^2+y^2=5$ at $(1,-2)$ also touches the circle $x^2+y^2-8x+6y+20=0$ at $P(\alpha ,\beta ),$ then the value of ${\alpha }^2+{\beta }^2$ is

Q. The equation of the tangent to the circle $x^2+y^2=5\mathrm{at}(1,-2)$ is x - 2y - 5 = 0.

1. True
2. False

Q. The equation of the tangent drawn to the circle $x^2+y^2=4$ at the point whose polar coordinates are given by $(2,\frac{\pi }{3})$ is

1. $x+\sqrt{3}y=2$
2. $\sqrt{3}x+y=2$
3. $\sqrt{3}x+y=4$
4. $x+\sqrt{3}y=4$

Q. The equation of the tangents to the circle $x^2+y^2=a^2$, which makes a triangle of area $a^2$ sq. units with coordinate axes, is/are

1. $x+y=a\sqrt{2}$
2. $x-y=a\sqrt{2}$
3. $2x+3y=2a\sqrt{3}$
4. $2x-3y=2a\sqrt{3}$

Q.

If a tangent drawn from the point (4, 0) to the circle $x^2+y^2=8$ touches it at a point A in the first quadrant, then the coordinates of another point B on the circle such that AB = 4 are

1. (-1, 1) or (1, -1)

2. (3, -2) or (-3, 2)

3. (2, -2) or (-2, 2)

4. (1, -2) or (-2, 1)

Q. Consider the relation $4l^2-5m^2+6l+1=0$, where $l,m\in R.$ If the line $lx+my+1=0$ touches a fixed circle, then the centre of that circle is:

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

Q. If $\alpha$ and $\alpha$ are two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and the chord joining these two points passes
through the focus (ae, 0) then e $cos\frac{\alpha -\beta }{2}$

1. 
2. 
3. 
4. 

Q. The area of the triangle formed by the tangent, normal at $P(1,1)$ on the curve $\sqrt{x}+\sqrt{y}=2$ with the $x$ -axis is

Q. Suppose tangent at a point $P$ on the circle $(x-4{)}^2+y^2=8$ is normal to the circle $x^2+(y-4{)}^2=4.$ Then the square of the distance of this point $P$ from the origin is

1. $16+24\sqrt{3}$
2. $32-16\sqrt{3}$
3. $64+32\sqrt{3}$
4. $16-8\sqrt{3}$

Q. If the variable line $y=kx+2h$ is tangent to an ellipse $2x^2+3y^2=6$ then locus of $P(h,k)$ is a conic $C$ whose eccentricity is $e$. Then $3e^2$ is

Q.

A circle of radius 2 has centre at (2, 0) another circle of radius 1 has centre at (5, 0). A line is tangent to the two circles at points in the quadrant. which of the following is the y-intercept of the line?

1. 3
2. $\frac{\sqrt{2}}{4}$
3. $\frac{8}{3}$
4. $2\sqrt{2}$
   

Q. The point of contact of the tangent to the circle $x^2+y^2=5$ at the point (1, -2) which touches the circle $x^2+y^2-8x+6y+20=0$, is

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

Q. If tangent to the circle $x^2+y^2=5$ at $(1,-2)$ also touches the circle $x^2+y^2-8x+6y+20=0$ at point $(h,k),$ then the value of $h+k$ is

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

Q. Given two circles x2+y2+5√2(x+y)=0 and x2+y2+7√2(x+y)=0. Let the radius of the third circle, which is tangent to the given circles and to their common diameter be 2P−1P then value of P/2 is

Q. A tangent drawn through the point $(2,-1)$ to a circle meets it at $(2,3)$. If radius of the circle is $3$ units, then equation of the circle can be

1. $x^2+y^2-10x-6y+25=0$
2. $x^2+y^2+2x-6y+1=0$
3. $x^2+y^2+2x+6y+1=0$
4. $x^2+y^2-6x-10y+19=0$

Q. The circle $C_1:x^2+y^2=3$, with centre at $O$, intersect the parabola $x^2=2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and $C_3$ have equal radii$\left(2\sqrt{3}\right)$ and centres $Q_2$ and $Q_3$ respectively. If $Q_2$ and $Q_3$ lie on the y-axis, then:

1. $Q_2{Q}_3=12$
2. $R_2{R}_3=4\sqrt{6}$
3. Area of the triangle $O{R}_2{R}_3$ is $6\sqrt{2}$
4. Area of the triangle $P{Q}_2{Q}_3$ is $4\sqrt{2}$

Q. Let $y=mx+c,m>0$ be the focal chord of $y^2=-64x,$ which is tangent to $(x+10{)}^2+y^2=4.$ Then the value of $4\sqrt{2}(m+c)$ is equal to

Q. If tangent to the circle $x^2+y^2=5$ at $(1,-2)$ also touches the circle $x^2+y^2-8x+6y+20=0$ at point $(h,k),$ then the value of $h+k$ is

Q. The equation of the tangents to the circle $x^2+y^2=a^2$, which makes a triangle of area $a^2$ sq. units with coordinate axes, is/are

1. $x+y=a\sqrt{2}$
2. $x-y=a\sqrt{2}$
3. $2x+3y=2a\sqrt{3}$
4. $2x-3y=2a\sqrt{3}$

Q. An equation of the tangent to the curve $y=x^4$ from the point (2, 0) not on the curve is

1. y=0
2. x + y=0
3. None of these
4. x =0

Q. The point of contact of the tangent to the circle $x^2+y^2=5$ at the point (1, -2) which touches the circle $x^2+y^2-8x+6y+20=0$, is

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

Q. If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is:

1. $x^2+y^2-2xy=0$
2. $x^2+y^2-2x^2{y}^2=0$
3. $x^2+y^2-4x^2{y}^2=0$
4. $x^2+y^2-16x^2{y}^2=0$

Q. Consider the relation $4l^2-5m^2+6l+1=0$, where $l,m\in R.$ If the line $lx+my+1=0$ touches a fixed circle, then the centre of that circle is:

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

Q. If the line x + y = 5 is a tangent to the circle $x^2+y^2-2x-4y+3=0$, then the coordinates of the point of contact are .

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

Q. A chord of the circle $x^2+y^2-4x-6y=0$ passing through the origin subtends an angle $ta{n}^{-1}\left(\frac{7}{4}\right)$ at the point where the circle meets positive y-axis. Equation of the chord is

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

Q. Consider a circle $x^2+y^2-2x-4y-20=0$ with centre $A$. If the tangents to the circle at points $B(1,7)\text{and}C(4,-2)$ meet at a point $D$, then the area of the quadrilateral $ABDC$ is

Q. Two tangents are drawn on the circle $x^2+y^2-6x-2y-15=0$ at point $B(3,6)$ and $D(0,-3)$ which meets at point $C$. If $A$ be the center of circle, then area of quadrilateral $ABCD$ is