Length of Tangent

Q.

If the curves $x=y^4$ and $\mathrm{xy}=\mathrm{k}$ cut at right angles, then $(4k{)}^6$ is equal to:

Q.

Find points on the curve $\frac{x^2}{9}+\frac{y^2}{16}=1$ at which the tangents are

(a) parallel to X-axis

(b) parallel to Y-axis.

Q. $P$ is a point on the positive $x-$axis, $Q$ is a point on the positive $y-$axis and $O$ is the origin. If the line passing through $P$ and $Q$ is tangent to the curve $y=3-x^2,$ then the minimum area of triangle $OPQ$ is

1. $16$ sq. units
2. $4$ sq. units
3. $1$ sq. unit
4. $2$ sq. units

Q.

Geometrically Rolle's theorem ensures that there is at least one point on the curve f(x) , whose abscissa lies in (a, b) at which the tangent is

1. Parallel to the line joining the end points of the curve

2. Parallel to the line y = x

3. Parallel to the y axis

4. Parallel to the x axis

Q. Find the equations of the tangent and normal to the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2\text{at}(1,1)$

Q. At any point $(x,y)$ of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point $(-4,-3)$.
Find the equation of the curve given that it passes through $(-2,1)$.

Q. A chord of the parabola $y=x^2-2x+5$ joins the point with the abscissas $x_1=1,x_2=3$. Then the equation of the tangent to the parabola parallel to the chord is :

1. $2x-y+\frac{5}{4}=0$
2. $2x-y+2=0$
3. $2x-y+1=0$
4. $2x+y+1=0$

Q. Find the equation of tangents to the curve $y=x^3+2x-4$, which are perpendicular to line $x+14y+3=0.$

Q.

In the above figure, length of tangent is length of PT

1. True

2. False

Q. If slope of tangent at atleast one point to the curve $y=x^3+3a{x}^2+3x+7$ is negative, then

1. $a\in (-\infty ,0)$
2. $a\in (-\infty ,-1)$
3. $a\in (-1,1)$
4. $a\in (1,\infty )$

Q.

Find the points on the curve $x^2+y^2-2x-3=0$ at which tangents are parallel to the X-axis.

Q.

Find points on the curve $\frac{x^2}{9}+\frac{y^2}{16}=1$ =1 at which the tangents are parallel to Y-axis.

Q. If the length of the tangent drawn at the point (1, 3) on the curve y = $3x^3$ is a, then the value of $9a^2$ is

1. 82
2. 81
3. 729

Q. At what point of the curve y = x² does the tangent make an angle of 45° with the x-axis?

Q. Let $P$ be any point on the curve $x^{2/3}+y^{2/3}=a^{2/3}$. Then, the length of the segment of the tangent between the coordinate axes of length.

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

Q. The order of the differential equation of all tangent lines to the parabola $y=x^2$ is

1. 3
2. 4
3. 2
4. 1

Q.

If the length of the tangent drawn at the point $(1,3)$ on the curve $y=3x^3$ is $a$, then find the value of $9a^2$

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Q. Equation of the tangent to the parabola $y^2=4x+5$ which is parallel to the line $y=2x+7$ is

1. $y=2x+3$
2. $y=2x-3$
3. $y=2x+5$
4. $y=2x5$

Q. Find the equation of the parabola the extremities of whose latus rectum are $(1,2)$ and $(1,-4)$.

Q. The number of different points on the curve $y^2=x(x+1{)}^2$, where the tangent to the curve drawn at (1, 2) meets the curve again, is

1. 0
2. 2
3. 3
4. 1

Q. A line bisecting the ordinate $PN$ of a point $P(a{t}^2,2at),t>0$ on the parabola $y^2=4ax$ is drawn parallel to the axis to meet the curve at $Q$. If $NQ$ meets the tangents at the vertex at the point $T$, then the coordinates of $T$ are

1. $(0,\frac{4}{3}at)$
2. $(0,2at)$
3. $(0,at)$
4. $(\frac{1}{4}a{t}^2,at)$

Q. Equation of the two tangents drawn from (1, 4) to the parabole $y^2=12x$ are

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

Q.

If the length of the tangent drawn at the point (1, 3) on the curve y = $3x^3$ is a, then find the value of $9a^2$

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Q. Tangent at any point on the curve $x^2{y}^3=a^5$ has an intercept between the axes. This intercept is divided in the ratio $m:n,$ such that $m>n$ and $m,n$ are co-prime, then $4m+2n=$

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

1. $e(x+y)=1$
2. $y+ex=1$
3. $x+y=e$
4. $x+ey=2$

Q. The equation of tangent to the curve $y=b{e}^{-x/a}$ where it cuts the $y$-axis is

1. $\frac{x}{a}+\frac{y}{b}=-1$
2. $\frac{x}{a}+\frac{y}{b}=1$
3. none of these
4. $\frac{x}{a}-\frac{y}{b}=1$

Q. The angle between the two tangents from the origin to the circle ${(x-7)}^2+{(y+1)}^2=25$ equals-

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. None of these.

Q.

If the length of the tangent drawn at the point (1, 3) on the curve y = $3x^3$ is a, then find the value of $9a^2$

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Q.

If the length of the tangent drawn at the point $(1,3)$ on the curve $y=3x^3$ is $a$, then find the value of $9a^2$

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Q. The equation of the parabola whose focus is the point $(0,0)$ and the tangent at the vertex is $x-y+1=0$ is

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