Slope Form of Tangent

Q.

What are the slope of $x-\mathrm{axis}$ and $y-\mathrm{axis}$?

Q.

Let $L$ be a tangent line to the parabola $y^2=4x–20$at $(6,2)$. If $L$ is also a tangent to the ellipse $\frac{x^2}{2}+\frac{y^2}{b}=1$, then the value of $b$ is equal to

1. $20$

2. $14$

3. $16$

4. $11$

Q.

If the tangent at $P(1,1)$ on $y^2=x{(2-x)}^2$ meets the curve again at $Q$, then $Q$ is:

1. $\left(-4,4\right)$

2. $\left(1,-2\right)$

3. $\left(\frac{9}{4},\frac{3}{8}\right)$

4. None of these.

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

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

Q.

A curve having the condition that the slope of the tangent at some point is two times the slope of the straight line joining the same point to the origin of coordinates is a/an

1. circle

2. ellipse

3. parabola

4. hyperbola

Q. Minimum area of circle which touches the parabolas $y=x^2+1$ and $y^2=x-1$ is

1. $\frac{9\pi }{16}$ squnits
2. $\frac{9\pi }{32}$ sq units
3. $\frac{9\pi }{8}$ squnits
4. $\frac{9\pi }{4}$ squnits

Q. Given: A circle, $2x^2+2y^2=5$ and a parabola, $y^2=4\sqrt{5}x$.
$\text{Statement-I}:$ An equation of a common tangent to these curves is $y=x+\sqrt{5}$
$\text{Statement-II}:$ If the line, $y=mx+\frac{\sqrt{5}}{m}(m\ne 0)$ is their common tangent, then $m$ satisfies $m^4-3m^2+2=0$.

1. Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
2. Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.
3. Statement-I is true; Statement-II is false.
4. Statement-I is false; Statement-II is true.

Q. Let $A(\sec \theta ,2\tan \theta )$ and $B(\sec \varphi ,2\tan \varphi )$, where $\theta +\varphi =\frac{\pi }{2},$ be two point on the hyperbola $2x^2-y^2=2.$ If $(\alpha ,\beta )$ is the point of the intersection of the normals to the hyperbola at $A$ and $B$, then $(2\beta {)}^2$ is equal to

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. The sum of the slopes of the tangents to the parabola $y^2=8x$ from the point $(-2,3),$ is

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

Q.

The length of tangent, subtangent, normal and subnormal for the curve $y=x^2+x-1$ at $\left(1,1\right)$ are $A,B,C$ and $D$ respectively, then their increasing order is

1. $B,D,A,C$

2. $B,A,C,D$

3. $A,B,C,D$

4. $B,A,D,C$

Q. The common tangent of the two parabolas $y^2=4x$ and $x^2=32y$ meets the coordinate axes at $A,B$ respectively. The equation of the circumcircle of $\Delta OAB$ is

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

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. $y=-30x-50$

Q. The equation of the tangent to the parabola $y^2=8x$ inclined at ${30}^{\circ }$ to the x axis is

1. 
2. 
3. 
4. 

Q. $\mathrm{If}\mathrm{the}\mathrm{parabolas}{y}^2=4x\mathrm{and}{x}^2=32y\mathrm{intersect}\mathrm{at}(16,8)\mathrm{at}\mathrm{an}\mathrm{angle}\theta ,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\theta \mathrm{is}$ .

1. ${\tan }^{-1}\left(\frac{3}{5}\right)$
2. ${\tan }^{-1}\left(\frac{5}{3}\right)$
3. ${\tan }^{-1}\left(\frac{4}{5}\right)$
4. ${\tan }^{-1}\left(\frac{5}{4}\right)$

Q. The slope of the line touching both the parabolas $y^2=4x$ and $x^2=-32y$ is :

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

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. The area bounded by the parabola $x=4-y^2$ and $y-$axis is:

1. $\frac{3}{32}$
2. $\frac{16}{4}$
3. $\frac{32}{3}$
4. $\frac{33}{2}$

Q. If the line $y=mx+c$ is tangent to the circle $x^2+y^2=5r^2$ and the parabola $y^2-4x-2y+4\lambda +1=0$ and point of contact of the tangent with the parabola is $(8,5),$ then the value of $(25r^2+\lambda +2m+c)$ is

Q. If a tangent to the parabola $y^2=8x$ meets the $x$-axis at $T$ and intersect the tangent at vertex $A$ at $P$, and the rectangle $TAPQ$ is completed, then the locus of the point $Q$ is

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

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. Find the equations of the tangent and the normal, to the curve $16x^2+9y^2=145$ at the point $(x_1,y_1)$, where $x_1=2$ and $y_1>0$.

Q.

What's the equation of tangent to the parabola $y^2=4ax$ having a slope 'm'.

1. $y=mx+a{m}^2$

2. $y=mx-a{m}^2$

3. $y=mx+\frac{a}{m}$

4. $y=mx-\frac{a}{m}$

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. The locus of a point, from where tangents to the rectangular hyperbola contain an angle of ${45}^0$, is

1. 
2. 
3. 
4. 

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

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 radius of the circle, which is touched by the line$y=x$ and has its center on the positive direction of the $x-$axis and also cuts off a chord of length$2$ units along the line $\sqrt{3}y-x=0$ is

1. $\sqrt{5}$

2. $\sqrt{3}$

3. $\sqrt{2}$

4. $1$

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