Length of Subnormal

Q.

The line $\frac{x}{a}+\frac{y}{b}=2$, touches the curve $\frac{x^n}{a^n}+\frac{y^n}{b^n}=2$, at

1. $(b,a)$

2. $(-b,-a)$

3. $(a,b)$

4. none of these

Q. ntShow that the tangents at the ends of a latus re tum of an ellipse intersect in the major axis.n

Q.

At any point of a curve (sub tangent) (sub normal) is equal to the square of the

Q. Find an equation of normal line to the curve y = x³ + 2x + 6 which is parallel to the line x + 14y + 4 = 0.

Q. Assertion :Any tangent to the curve $y=x^7+8x^3+2x+1$ makes an acute angle with the positive x - axis Reason: Any tangent to the curve $y=a_0{x}^{2n-1}+a_1{x}^{2n-1}+a_2{x}^{2n-3}+.......a_{n}x+1$ makes an acute angle with the positive x - axis where $a_1,........a_{n-1}\ge 0;a_0>0$ and $n\in N$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q.

In the figure, length of subnormal is the length P₁N (tangent and normal is drawn at the point P)

1. False

2. True

Q. If ${\left(\frac{a}{b}\right)}^{\frac{1}{3}}+{\left(\frac{b}{a}\right)}^{\frac{1}{3}}=4$, then the acute angle $\left(\theta \right)$ of intersection of the parabolas $y^2=4ax$ and $x^2=4by$ at a point other than the origin is

1. $\theta ={\tan }^{-1}\frac{3}{8}$
2. $\theta ={\tan }^{-1}\frac{5}{8}$
3. $\theta ={\tan }^{-1}\frac{3}{7}$
4. $\theta ={\tan }^{-1}1$

Q. If ST and SN are the lengths of the subtangent and the subnormal at the point $\theta =\frac{\pi }{2}$ on the curve $x=a(\theta +sin\theta ),y=at(1-cos\theta ),a\ne 1,$ then
[Karnataka CET 2005]

1. ST = SN
2. ST = 2 SN
3. $S{T}^2=aS{N}^3$
4. $S{T}^3=aSN$

Q. The tangent at any point (x, y) of a curve makes an angle tan⁻¹(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).

Q. The length of the sub-tangent and sub-normal is equal for the curve $y=e^{\mathrm{px}}+\mathrm{px}$ at the point (0, 1), then the value of p is

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

Q. Write the equation of the normal to the curve y = cos x at (0, 1).

Q. If the length of subnormal is four times the length of subtangent at a point $(3,4)$ on the curve $y=f\left(x\right)$. The tangent at $(3,4)$ to $y=f\left(x\right)$ meets the coordinate axes at $P$ and $Q$ and the maximum area of triangle $OPQ$ (where $O$ is origin) is $4b^2$, then $b=$

1. $\pm 2$
2. $\pm \frac{5}{2}$
3. $\pm \frac{3}{2}$
4. $\pm \frac{1}{2}$

Q. If the length of subnormal is four times the length of subtangent at a point $(3,4)$ on the curve $y=f\left(x\right)$. The tangent at $(3,4)$ to $y=f\left(x\right)$ meets the coordinate axes at $P$ and $Q$ and the maximum area of triangle $OPQ$ (where $O$ is origin) is $4b^2$, then $b=$

1. $\pm 2$
2. $\pm \frac{5}{2}$
3. $\pm \frac{3}{2}$
4. $\pm \frac{1}{2}$

Q. The length of the sub-tangent and sub-normal is equal for the curve $y=e^{\mathrm{px}}+\mathrm{px}$ at the point (0, 1), then the value of p is

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

Q. Find the slopes of the tangents of the curve $y=(x+1)(x-3)$ at the points where it cuts the X-axis.

1. $4$
2. $-4$
3. $2$
4. $-2$

Q. Find the equations of the tangent and the normal to the following curves at the indicated points.

(i) y = x⁴ − bx³ + 13x² − 10x + 5 at (0, 5) [NCERT]
(ii) y = x⁴ − 6x³ + 13x² − 10x + 5 at x = 1 [NCERT, CBSE 2011]
(iii) y = x² at (0, 0) [NCERT]
(iv) y = 2x² − 3x − 1 at (1, −2)
(v) $y^2=\frac{x^3}{4-x}\mathrm{at}\left(2,-2\right)$
(vi) y = x² + 4x + 1 at x = 3 [CBSE 2004]
(vii) $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\mathrm{at}\left(a\cos \theta ,b\sin \theta \right)$
(viii) $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\mathrm{at}\left(a\sec \theta ,b\tan \theta \right)$
(ix) y² = 4ax at $\left(\frac{a}{m^2},\frac{2a}{m}\right)$
(x) $c^2\left(x^2+y^2\right)=x^2{y}^2\mathrm{at}\left(\frac{c}{\cos \theta },\frac{c}{\sin \theta }\right)$
(ix) xy = c² at $\left(ct,\frac{c}{t}\right)$
(xii) $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\mathrm{at}\left(x_1,y_1\right)$
(xiii) $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\mathrm{at}\left(x_0,y_0\right)$ [NCERT]
(xiv) $x^{\frac{2}{3}}+y^{\frac{2}{3}}$ = 2 at (1, 1) [NCERT]
(xv) x² = 4y at (2, 1)
(xvi) y² = 4x at (1, 2) [NCERT]
(xvii) 4x² + 9y² = 36 at (3cosθ, 2sinθ) [CBSE 2011]
(xviii) y² = 4ax at (x₁, y₁) [CBSE 2012]
(xix) $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\mathrm{at}\left(\sqrt{2}a,b\right)$ [CBSE 2014]

Q. The number of tangents to the curve $x^{3/2}+y^{3/2}=2a^{3/2}$, $a>0$, which are equally inclined to the axes, is

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

Q. Find the length of subnormal at x= 2 on the curve y = $x^3$.

1. 96

Q.

Find the length of subnormal at x= 2 on the curve y = $x^3$.

___

Q.

In the figure, length of subnormal is the length P1N (tangent and normal is drawn at the point P)

1. T

Q. In the curve $y={\mathrm{ce}}^{\frac{x}{a}}$, the

1. The length of a sub-tangent is constant
2. The length of a sub-normal varies as the square of the ordinate
3. tangent at $(x_1,y_1)$ on the curve intersects the x-axis at a distance of $(x_1-a)$ from the origin.
4. equation of the normal at the point where the curve cuts y-axis is $\mathrm{cy}+\mathrm{ax}=c^2$

Q.

Find the length of subnormal at x= 2 on the curve y = $x^3$.

___

Q. Find the length of subnormal at x= 2 on the curve y = $x^3$.

1. 96

Q. In the curve $y={\mathrm{ce}}^{\frac{x}{a}}$, the

1. The length of a sub-tangent is constant
2. The length of a sub-normal varies as the square of the ordinate
3. tangent at $(x_1,y_1)$ on the curve intersects the x-axis at a distance of $(x_1-a)$ from the origin.
4. equation of the normal at the point where the curve cuts y-axis is $\mathrm{cy}+\mathrm{ax}=c^2$

Q. Find the slope of the tangent to the curve $y=x^3-3x+2$ at the point whose $x$ coordinate is $3$.

Q. Find the equations of all lines of slope zero and that are tangent to the curve $y=\frac{1}{x^2-2x+3}$.

Q. The angle between the tangents drawn from the point $(-a,2a)$ to $y^2=4ax$ is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{6}$

Q. If ST and SN are the lengths of the subtangent and the subnormal at the point $\theta =\frac{\pi }{2}$ on the curve $x=a(\theta +sin\theta ),y=at(1-cos\theta ),a\ne 1,$ then
[Karnataka CET 2005]

1. $S{T}^2=aS{N}^3$
2. $S{T}^3=aSN$
3. ST = SN
4. ST = 2 SN

Q. The point on the curve $y=x^2+4$ at which the tangent is perpendicular to the line $x+2y=3$ is

1. $(5,1)$
2. $(-1,5)$
3. $(1,-5)$
4. $(1,5)$

Q. If the slope of the tangent to the curve $y=x^3$ at a point on it is equal to the ordinate of the point then the point is

1. $(27,3)$
2. $(3,27)$
3. $(3,3)$
4. $(1,1)$