Slope Form of Tangent

Q. The equation of the common tangent touching the circle $(x-3{)}^2+y^2=9$ and the parabola $y^2=4x$ above the X-axis is

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

Q. The equation of a tangent to the parabola, $x^2=8y$, which makes an angle $\theta$ with the positive direction of $x$-axis, is :

1. $y=x\tan \theta -2\cot \theta$
2. $x=y\cot \theta +2\tan \theta$
3. $x=y\cot \theta -2\tan \theta$
4. $y=x\tan \theta +2\cot \theta$

Q.

The intercepts on $x-$axis, made by tangents to the curve, $y={\int }_0^x\left|t\right|dt,x\in \mathrm{ℝ}$ which are parallel to the line $y=2x$, are

1. $\pm 1$

2. $\pm 2$

3. $\pm 3$

4. $\pm 4$

Q. If $y=mx+c$ touches the parabola $y^2=4a(x+a)$, then

1. $c=\frac{a}{m}$
2. $c=am+\frac{a}{m}$
3. $c=a+\frac{a}{m}$
4. $\text{none of these}$

Q.

The coordinates of the point of contact of the tangent to the parabola $y^2=16x$, which is perpendicular to the line $2x-y+5=0$ are

1. (16, 16)

2. (16, -16)

3. (1, 4)

4. (1, -4)

Q.

The equation of the tangent to the curve $y=e^{-\left|x\right|}$ at the point where the curve cuts the line $x=1$ is

1. $e(x+y)=1$

2. $y=e^x+1$

3. $x+y=e$

4. None of these

Q. The locus point of intersection of tangents to the parabola $y^2=4ax$, the angle between them being always ${45}^{\circ }$
is

1. $x^2-y^2+6ax-a^2=0$
2. $x^2-y^2-6ax+a^2=0$
3. $x^2-y^2+6ax+a^2=0$
4. $x^2-y^2-6ax-a^2=0$

Q. A tangent is drawn to the parabola $y^2=4x$ at a point $P$ on the parabola in the first quadrant and another tangent is drawn to the vertex $A$ of the parabola. Let both the tangent meet at a point $B$, if area of the triangle $ABP=32{\text{unit}}^2$, then equation of the tangent is

1. $4y=x+8$
2. $2y=x+8$
3. $4y=x+16$
4. $4x=y+16$

Q. The sum of $x-$intercept and $y-$intercept of the common tangent to the parabola $y^2=16x$ and $x^2=128y$ is

1. $-16$
2. $-32$
3. $-8$
4. $-24$

Q. Two tangents are drawn to end points of the latus rectum of the parabola $y^2=4x$. The equation of the parabola which touches both the tangents as well as the latus rectum is

1. $y^2=8x$
2. $y^2=8(x-1)$
3. $y^2=4(x-1)$
4. $(y-1{)}^2=8x$

Q. The equation of the tangent to the parabola $y^2=16x$ inclined at an angle of ${60}^{\circ }$ to the positive $x-$axis is

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

Q.

If the line $ky-2x-k^2+2h=0\&$ parabola $x^2=4y$ touches each other, then

1. $\frac{k}{2}(k^2+2h)=2$

2. $\frac{k}{4}(k^2+2h)=4$

3. $\frac{k}{4}(-k^2+2h)=2$

4. $\frac{k}{4}(-k^2+2h)=4$

Q. If line $y=2x+\frac{1}{4}$ is tangent to $y^2=4ax$ then a is equal to

1. $\frac{1}{2}$
2. $1$
3. $2$
4. $Noneofthese$

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

The tangent to a suitable conic (Column 1) at $(\sqrt{3},\frac{1}{2})$ is found to be $\sqrt{3}x+2y=4$, then which of the following options is the only CORRECT combination?

1. $\left(IV\right)\left(iii\right)\left(S\right)$
2. $\left(IV\right)\left(iv\right)\left(S\right)$
3. $\left(II\right)\left(iii\right)\left(R\right)$
4. $\left(II\right)\left(iv\right)\left(R\right)$

Q. Two straight lines are perpendicular to each other.One of them touches the parabola $y^2=4a(x+a)$ and the other touches $y^2=4b(x+b)$ .The locus of the point of intersection of the two lines is

1. x + a = 0
2. x + b = 0
3. x + a + b = 0
4. x – a – b = 0

Q. Equation of common tangent of $y=x^2,y=-x^2+4x-4$ is

1. $y=4(x-1)$
2. $y=0$
3. $y=-4(x-1)$
4. $y=-30x-50$