T

Q. If the eccentricity of the standard hyperbola passing through the point $(4,6)$ is $2$, then the equation of the tangent to the hyperbola at $(4,6)$ is :

1. $3x-2y=0$
2. $2x-y-2=0$
3. $2x-3y+10=0$
4. $x-2y+8=0$

Q. If the common tangent to the parabolas, $y^2=4x$ and $x^2=4y$ also touches the circle, $x^2+y^2=c^2$, then $c$ is equal to:

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

Q. The locus of the mid points of the chords of the hyperbola $x^2-y^2=4,$ which touch the parabola $y^2=8x,$ is

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

Q. If a line along a chord of the circle $4x^2+4y^2+120x+675=0,$ passes through the point $(-30,0)$ and is tangent to the parabola $y^2=30x,$ then the length of this chord is

1. $3\sqrt{5}$
2. $5\sqrt{3}$
3. $7$
4. $5$

Q. A tangent line $L$ is drawn at the point $(2,-4)$ on the parabola $y^2=8x$. If the line $L$ is also tangent to the circle $x^2+y^2=a,$ then $a$ is equal to

Q.

If the tangent at point $P$ on the circle $x^2+y^2+6x+6y-2=0$ meets the straight line $5x-2y+6=0$ at a point $Q$ on $y-axis$, the length of $PQ$ is:

1. $4$

2. $2\sqrt{5}$

3. $5$

4. $3\sqrt{5}$

Q. For the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ distance between the foci is $10.$ From the point $(2,\sqrt{3}),$ perpendicular tangents are drawn to the hyperbola. If eccentricity of the hyperbola is $e,$ then the value of $4e$ is

Q. The tangent to the parabola $y^2=4x$ at the point where it intersects the circle $x^2+y^2=5$ in the first quadrant, passes through the point :

1. $(-\frac{1}{3},\frac{4}{3})$
2. $(-\frac{1}{4},\frac{1}{2})$
3. $(\frac{3}{4},\frac{7}{4})$
4. $(\frac{1}{4},\frac{3}{4})$

Q. Equation of the tangent to $y^2=8x$ which is parallel to x-y+3=0 is

1. x-y+4=0

2. x-y+5=0

3. x-y+2=0

4. x-y+7=0

Q.

The sum of intercepts on coordinate axes made by tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ is

1. $a$

2. $2a$

3. $2\sqrt{a}$

4. None of these

Q.

Find the equation of tangent to parabola $y^2=4axat(a{t}^2,2at)$

1. $ty=x+a{t}^2$

2. $y=tx+a{t}^2$

3. $ya{t}^2=x+t$

4. $y+tx=a{t}^2$

Q.

Lines are drawn to the point $P(-2,-3)$to meet the circle $x^2+y^2-2x-10y+1=0$. The length of the line segment $PA$, $A$being the point on the circle where the line meets the circle at coincident is,

1. $16$

2. $4\sqrt{3}$

3. $48$

4. None of these

Q. Let $L_1$ be a tangent to the parabola $y^2=4(x+1)$ and $L_2$ be a tangent to the parabola $y^2=8(x+2)$ such that $L_1$ and $L_2$ intersect at right angles. Then $L_1$ and $L_2$ meet on the straight line:

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

Q.

A tangent to the parabola $y^2=4ax$ meets x axis at a point T. This tangent meets the tangent at the vertex at point P. If rectangle TAPQ is completed then the locus of Q is.

1. Ellipse

2. Circle

3. Straight line

4. parabola

Q. If the chords of tangents from two points $(–4,2)$ and $(2,1)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are at right angle, then the eccentricity of the hyperbola is

1. $\sqrt{2}$
2. $\frac{\sqrt{7}}{2}$
3. $\sqrt{\frac{3}{2}}$
4. $\sqrt{\frac{5}{3}}$

Q.

A hyperbola passes through the point $P(\sqrt{2},\sqrt{3})$and has foci at $(\pm 2,0)$.

Then the tangent to this hyperbola at $P$ also passes through the point

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

2. $(\sqrt{3},\sqrt{2})$

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

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

Q.

The equation of the tangent at $(x_1,y_1)$ to a curve $y^2=4ax$ at a point $(x_1,y_1)$ is given by $y{y}_1=2a(x+x_1)$

1. True

2. False

Q. Let $PQ$ be a focal chord of the parabola $y^2=4ax.$ The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,$ $a>0.$ Length of chord $PQ$ is

1. $7a$
2. $5a$
3. $2a$
4. $3a$

Q. The equation of the tangent to the curve $y=2t^2+t,x=t^2$ at $(1,3)$ is

1. $y=x+2$
2. $2y=x+5$
3. $2y=3x+3$
4. $2y=5x+1$

Q. The equation of the common tangent of the parabola $y^2=8x$ and $x^2+12y^2=48$ is

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

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

1. x + y = e
2. e(x + y) = 1
3. y + ex = 1
4. none of these

Q. What is the angle between the tangents to the curve $y=x^2-5x+6$ at the points (2, 0) and (3, 0)

1. 

2. 

3. 

4. 

Q.

Find the equation of tangent to parabola $y^2=4axat(a{t}^2,2at)$

1. 

2. 

3. 

4. 

Q. The equation of the common tangent to the equal parabolas $y^2=4ax$ and $x^2=4ay$ is

1. $x+y+a=0$
2. $x+y=a$
3. $x-y=a$
4. none of these

Q. The tangent to the parabola $y^2=4x$ at the point where it intersects the circle $x^2+y^2=5$ in the first quadrant, passes through the point :

1. $(-\frac{1}{3},\frac{4}{3})$
2. $(-\frac{1}{4},\frac{1}{2})$
3. $(\frac{3}{4},\frac{7}{4})$
4. $(\frac{1}{4},\frac{3}{4})$

Q. The equation of tangent to the ellipse $x^2+4y^2=5$ at (-1, 1), is

1. $x+4y-5=0$
2. $x+4y+5=0$
3. $x-4y+5=0$
4. $x-4y-5=0$

Q.

A tangent to the parabola $y^2=4ax$ meets x axis at a point T. This tangent meets the tangent at the vertex at point P. If rectangle TAPQ is completed then the locus of Q is.

1. Circle

2. Straight line

3. parabola

4. Ellipse

Q. Find the equation of tangent to the curve
$y={\sin }^{-1}\frac{2x}{1+x^2}atx=\sqrt{3}$

Q. Write the equation of a tangent to the graphs of the following curves at the indicated points
$y=x^2{e}^{-x}$ at the point x = 1.

Q. Find the equation of common tangents to the two parabolas $y^2=32x$ and $x^2=108y$