Equation of Tangent in Slope Form

Q.

If the tangent at $\left(1,7\right)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16x+12y+c=0$, then the value of $c$ is:

1. $85$

2. $95$

3. $195$

4. $185$

Q.

The equation of the tangents to the circle $x^2+y^2=13$ at the point whose abscissa is $2$, are

1. $2x+3y=13,2x–3y=13$

2. $3x+2y=13,2x–3y=13$

3. $2x+3y=13,3x–2y=13$

4. None of these above

Q.

The equation of circle with centre $(-a,-b)$ and radius $\sqrt{a^2-b^2}$is

1. $x^2+y^2+2ax+2by+2b^2=0$

2. $x^2+y^2–2ax–2by–2b^2=0$

3. $x^2+y^2–2ax–2by+2b^2=0$

4. $x^2+y^2–2ax+2by+2a^2=0$

Q.

The circle $C_1:x^2+y^2=3$ having centre at origin, O intersects 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 $2\sqrt{3}$ 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 $\Delta O{R}_2{R}_{3}is6\sqrt{2}$

4. area of the $\Delta P{Q}_2{Q}_{3}is4\sqrt{2}$

Q. Equation of a common tangent to the circle, $x^2+y^2-6x=0$ and the parabola, $y^2=4x$, is

1. $2\sqrt{3}y=-x-12$
2. $2\sqrt{3}y=12x+1$
3. $\sqrt{3}y=3x+1$
4. $\sqrt{3}y=x+3$

Q.

The set of values of 'c' so that the equations $y=\left|x\right|+cand{x}^2+y^2-8\left|x\right|-9=0$ have no solution is

1. $(\infty ,-3)\cup (3,\infty )$
   

2. $(-3,3)$
   

3. $(-\infty ,5\sqrt{2})\cup \left(5\sqrt{(}2\right),\infty )$
   

4. $5\sqrt{2}-4,\infty$

Q. The product of slope of tangents from point $(0,1)$ to circle $x^2+y^2-2x+4y=0$ is:

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

Q. The area (in sq. units) of the smaller of the two circles that touch the parabola, $y^2=4x$ at the point $(1,2)$ and the $x$-axis is :

1. $4\pi (3+\sqrt{2})$
2. $8\pi (2-\sqrt{2})$
3. $8\pi (3-2\sqrt{2})$
4. $4\pi (2-\sqrt{2})$

Q. If a line $y=mx+c$ is a tangent to the circle $(x-3{)}^2+y^2=1$ and it is perpendicular to a line $L_1,$ where $L_1$ is the tangent to the circle $x^2+y^2=1$ at the point $(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}),$ then

1. $c^2+7c+6=0$
2. $c^2-6c+7=0$
3. $c^2+6c+7=0$
4. $c^2-7c+6=0$

Q. If a line $y=mx+c$ is a tangent to the circle $(x-3{)}^2+y^2=1$ and it is perpendicular to a line $L_1$, where $L_1$ is the tangent to the circle $x^2+y^2=1$ at the point $(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$; then:

1. $c^2+7c+6=0$
2. $c^2-6c+7=0$
3. $c^2-7c+6=0$
4. $c^2+6c+7=0$

Q. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x-5y=20$ to the circle $x^2+y^2=9$ is

1. $20(x^2+y^2)-36x+45y=0$
2. $20(x^2+y^2)+36x-45y=0$
3. $36(x^2+y^2)-20y+45y=0$
4. $36(x^2+y^2)+20x-5y=0$

Q. If the tangent drawn at $(1,2)$ to the circle $x^2+y^2-4x+2y-5=0$ is normal to the circle $x^2+y^2-4x+ky-1=0$, then the value of $|3k|$ is

Q. If the straight line $y=x+4$ cuts the circle $x^2+y^2=26$ at $P$ and $Q$, where $Q$ is in first quadrant, then which of the following is/are correct?

1. The equation of tangent at $P$ is $5x+y+26=0$
2. The equation of tangent at $Q$ is $x+5y-26=0$
3. The point of intersection of tangents is $(-\frac{13}{2},\frac{13}{2})$
4. The mid-point of chord $PQ$ is $(2,-2)$

Q. A tangent PT is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1)$. A straight line L, perpendicular to PT is a tangent to the circle $(x-3{)}^2+y^2=1$
A common tangent of the two circles is

1. $x=4$
2. $y=2$
3. $x+\sqrt{3}y=4$
4. $x+2\sqrt{2}y=6$

Q. The equations of the tangents to the circle $x^2+y^2$ - 4x - 6y - 12 = 0 and parallel to 4x - 3y = 1 are

1. 4x + 3y + 14 = 0, 4x + 3y + 16 = 0
2. 4x – 3y – 24 = 0, 4x – 3y + 26 = 0
3. x – y – 14 = 0, x – y + 16 = 0
4. 4x – 3y + 34 = 0, 4x – 3y + 16 = 0

Q. Consider a circle $x^2+y^2+ax+by+c=0$ lying completely in the first quadrant. If $m_1$ and $m_2$ are the maximum and the minimum values of $\frac{y}{x}$ for all ordered pairs $(x,y)$ on the circumference of the circle, then the value of $(m_1+m_2)$ is

1. $\frac{2ab}{4c-b^2}$
2. $\frac{2ab}{b^2-4ac}$
3. $\frac{2ab}{b^2-4c}$
4. $\frac{2ab}{4ac-b^2}$

Q. The equations of the tangents to the circle $x^2+y^2-6x+4y=12$ which are parallel to the straight line 4x + 3y + 5 = 0, are

1. 3x - 4y - 19 = 0, 3x - 4y + 31 = 0
2. 4x + 3y - 19 = 0, 4x + 3y + 31 = 0
3. 4x + 3y + 19 = 0, 4x + 3y - 31 = 0
4. 3x - 4y + 19 = 0, 3x - 4y + 31 = 0

Q. The length of the chord, which is tangent to the circle $x^2+y^2=a^2$ with respect to its director circle is

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

Q. The equation of tangent parallel to the line $y=x$ drawn to the circle $x^2+y^2=9$ is/are

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

Q. If $y=x$ be the tangent to the circle $x^2+y^2+2gx+2fy+c=0$ at ponit $P$ such that the distance of $P$ from origin is $4\sqrt{2},$ then the value of $c$ is

Q. The slope m of a tangent through the point (7, 1) to the circle $x^2+y^2=25$ satisfies the equation

1. $12m^2$ + 7m - 12 = 0
2. $12m^2$ + 7m + 9 = 0
3. $12m^2$ - 7m - 12 = 0
4. $9m^2$ + 12m + 16 = 0

Q.

A tangent PT is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1)$. A straight line L, perpendicular to PT is a tangent to the circle $(x-3{)}^2+y^2=1$
A possible equation of L is

1. $x-\sqrt{3}y=1$
2. $x+\sqrt{3}y=1$
3. $x-\sqrt{3}y=-1$
4. $x+\sqrt{3}y=5$

Q. A circle with centre $(a,b)$ passes through the origin. The equation of the tangent to the circle at the origin is

1. $ax-by=0$
2. $ax+by=0$
3. $bx-ay=0$
4. $bx+ay=0$

Q. If the straight line $y=x+4$ cuts the circle $x^2+y^2=26$ at $P$ and $Q$, where $Q$ is in first quadrant, then which of the following is/are correct?

Q.

If O is the origin and OP, OQ are distinct tangents to the circle $x^2+y^2+2gx+2fy+c=0,$ the circumcentre of the triangle OPQ is

1. (-g, -f)
2. (g, f)
3. (-f, -g)
4. None of these

Q. A circle $x^2+y^2+4x-2\sqrt{2}y+c=0$ is the director circle of circle $S_1$ and $S_1$ is the director circle of circle $S_2$ and so on. If the sum of radii of all these circles is $2$ units and the value of $c$ is $4\sqrt{k},$ then the value of $k$ is

Q. Let tangents are drawn at two points of the circle $(x-7{)}^2+(y+1{)}^2=25$. If the point of intersection of both the tangents is origin, then the angle between them (in degrees) is

Q. Two tangents are drawn from $(1,-2)$ to the circle $x^2+y^2-8x+6y+20=0.$ Which of the following is false ?

1. angle between the tangents is $\frac{\pi }{2}$
2. equation of one of the two tangents is $x-2y-5=0$
3. one of the two tangents passes through the origin
4. sum of the squares of the slopes of the two tangents is $\frac{15}{4}$