Equation of Tangent in Slope Form

Q. If the points $(2,0),(0,1),(4,0)$ and $(0,a)$ are concyclic then $a=$

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

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 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. Let a parabola $P$ be such that its vertex and focus lie on the positive $x$-axis at a distance $2$ and $4$ units from the origin, respectively. If tangents are drawn from $O(0,0)$ to the parabola $P$ which meet $P$ at $S$ and $R$, then the area (in sq. units) of $\Delta SOR$ is equal to

1. $16$
2. $32$
3. $16\sqrt{2}$
4. $8\sqrt{2}$

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

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. The tangent to the circle $C_1:x^2+y^2-2x-1=0$ at the point $(2,1)$ cuts off a chord of length $4$ from a circle $C_2$ whose centre is $(3,-2)$. The radius of $C_2$ is :

1. $\sqrt{6}$
2. $3$
3. $\sqrt{2}$
4. $2$

Q. The value of $\lambda$ for which the set $\left\{\right(x,y):x^2+y^2-6x+4y\le 12\}\cap \left\{\right(x,y):4x+3y\le \lambda \}$ contains only one point is

1. $19$
2. $-31$
3. $31$
4. $-19$

Q. Equation of the tangent to the circle, at the point $(1,–1)$, whose centre is the point of intersection of the straight lines $x–y=1$ and $2x+y=3$ is :

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

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. If two tangents drawn from a point $(\alpha ,\beta )$ lying on the ellipse $25x^2+4y^2=1$ to the parabola $y^2=4x$ are such that the slope of one tangent is four times the other, then the value of $(10\alpha +5{)}^2+(16{\beta }^2$ $+50{)}^2$ equals

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. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
​​​​​​$\begin{array}{ccc}Column1& Column2& Column3\\ \left(I\right)x^2+y^2=a^2& \left(i\right)my=m^{2}x+a& \left(P\right)(\frac{a}{m^2},\frac{2a}{m})\\ \left(II\right)x^2+a^2{y}^2=a^2& \left(ii\right)y=mx+a\sqrt{m^2+1}& \left(Q\right)(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}})\\ \left(III\right)y^2=4ax& \left(iii\right)y=mx+\sqrt{a^2{m}^2-1}& \left(R\right)(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}})\\ \left(IV\right)x^2-a^2{y}^2=a^2& \left(iv\right)y=mx+\sqrt{a^2{m}^2+1}& \left(S\right)(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}})\end{array}$

For $a=\sqrt{2}$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact $(-1,1)$, then which of the following options is the only CORRECT combination for obtaining its equation?

1. $\left(I\right)\left(i\right)\left(P\right)$
2. $\left(I\right)\left(ii\right)\left(Q\right)$
3. $\left(II\right)\left(ii\right)\left(Q\right)$
4. $\left(III\right)\left(i\right)\left(P\right)$

Q. The equation of the tangents to the circle $x^2+y^2=9$ which make an angle of $\frac{\pi }{3}$ with the $x$-axis

1. $x-\sqrt{3}y-6=0$
2. $x-\sqrt{3}y+6=0$
3. $\sqrt{3}x-y+6=0$
4. $\sqrt{3}x-y-6=0$

Q. The line lx + my + n = 0 will be a tangent to the circle $x^2+y^2=a^2$ iff

1. $n^2(l^2+m^2)=a^2$
2. $a^2(l^2+m^2)=n^2$
3. n(l + m) = a
4. a(l + m) = n

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 $B$. What is the length of the chord $AB$?

Q. If $\forall h\in R-\left\{0\right\},$ two distinct tangents can be drawn from the points $(2+h,3h-1)$ to the curve $y=x^3-6x^2-a+bx$, then $\frac{a}{b}$ is

Q. From the point $P(\alpha ,\beta )$, tangents are drawn to the parabola $y^2=4x$, including an angle ${45}^{\circ }$ to each other. Then locus of $P(\alpha ,\beta )$ is

1. a circle with centre $(–3,0)$
2. an ellipse with centre $(–3,0)$
3. a rectangular hypererbola with centre $(–3,0)$
4. a rectangular hypererbola with centre $(3,0)$

Q. Tangents to the parabola $y^2=4x$ at $P$ and $Q$ meet at $T(x_1,0)$ and normals at $P$ and $Q$ meet at $R(x_2,0).$ If $x_2$ is the length of the latus rectum of the ellipse $3x^2+8y^2=48,$ then the area of the quadrilateral $PTQR$ is

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. 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. 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 $\frac{15}{4}$

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

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. 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. 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.

Find the area of the region in the first quadrant enclosed by X-axis and $x=\sqrt{3}$ y and the circle $x^2+y^2=4$

Q. The point of intersection of the tangent $x+\sqrt{3}y=8$ with the circle $x^2+y^2=16$ is

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