Equation of Tangent at a Point (x,y) in Terms of f'(x)

Q.

Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+m–n=0$ and $l^2+m^2–n^2=0$. Then the value of ${\sin }^4\alpha +{\cos }^4\alpha$ is:

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

2. $\frac{1}{2}$

3. $\frac{5}{8}$

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

Q.

If the tangents on the ellipse $4x^2+y^2=8$ at the points $(1,2)\mathrm{and}(a,b)$ are perpendicular to each other, then $a^2$ is equal to:

1. $\frac{2}{17}$

2. $\frac{64}{17}$

3. $\frac{128}{17}$

4. $\frac{4}{17}$

Q. The sum of intercepts on co-ordinate axes made by tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ is

1. a
2. 2a
3. None of these
4. $2\sqrt{a}$

Q.

Coordinates of a point on the curve $y=x\log \left(x\right)$ at which the normal is parallel to the line $2x-2y=3$ are

1. $\left(0,0\right)$

2. $\left(e,e\right)$

3. $\left(e^2,2e^2\right)$

4. $\left(e^{-2},-2e^{-2}\right)$

Q. The tangent to the curve given by, $x=e^t.\cos t,y=e^t.\sin t\mathrm{at}t=\frac{\pi }{4},$ makes an angle of with x-axis.

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

Q. The angle between curves $y^2=4x$ and $x^2+y^2=5$ at (1, 2) is

1. $ta{n}^{-1}\left(3\right)$
2. $ta{n}^{-1}\left(2\right)$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{4}$

Q.

The angle between the lines represented by the equation $x^2-2pxy+y^2=0$ , is

1. ${\sec }^{-1}\left(p\right)$

2. ${\cos }^{-1}\left(p\right)$

3. ${\tan }^{-1}\left(p\right)$

4. None of these

Q.

If the normal to the curve$y^2=5x–1$, at the point $(1,–2)$ is of the form$ax–5y+b=0$, then $a$ and $b$ are

1. $4,–14$

2. $4,14$

3. $–4,14$

4. $–4,–14$

Q.

The abscissae of the points, where the tangent to curve $y=x^3-3x^2-9x+5$ is parallel to $x$-axis are

1. $x=0\&0$

2. $x=1\&-1$

3. $x=1\&-3$

4. $x=-1\&3$

Q. The tangent to the curve $y=x^2-5x+5$, parallel to the line $2y=4x+1$, also passes through the point :

1. $(\frac{7}{2},\frac{1}{4})$
2. $(\frac{1}{4},\frac{7}{2})$
3. $(-\frac{1}{8},7)$
4. $(\frac{1}{8},-7)$

Q. If a variable tangent to the curve $x^{2}y=c^3$ makes intercepts $a,b$ on $x$ and $y-$axis respectively, then the value of $a^{2}b$ is

1. $\frac{4}{27}{c}^3$
2. $\frac{27}{4}{c}^3$
3. $\frac{4}{9}{c}^3$
4. $27c^3$

Q.

Find the foot of perpendicular from the point (2, 3, -8) to $\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$. Also, find the perpendicular distance from the given point to the line.

Q.

The equation of the normal to the curve $y={(1+x)}^y+{\sin }^{-1}\left({\sin }^2\left(x\right)\right)$ at $x=0$.

1. $x+y=2$

2. $x+y=1$

3. $x–y=1$

4. $x^2–y^2=2$

Q.

The area of the region bounded by the parabola ${(y-2)}^2=x-1$, the tangent to the parabola at the point $(2,3)$ and the $x$-axis is

1. $6\mathrm{sq}$ units

2. $9\mathrm{sq}$ units

3. $12\mathrm{sq}$ units

4. $3\mathrm{sq}$ units

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

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

Q.

If $x=t^2$ and $y=2t$, then equation of the normal at $t=1$ is:

1. $x+y-3=0$

2. $x+y-1=0$

3. $x+y+1=0$

4. $x+y+3=0$

Q.

The tangent to the curve $x^2+y^2=25$ parallel to the line 3x-4y=7 exist at the point

1. (-3, -4)

2. (3, -4)

3. (3, 4)

4. None of these

Q.

If the tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ at any point on it cuts the axes OX and OY at P and Q respectively, then OP +OQ is

1. a

2. 4a

3. 2a

4. a/2

Q.

If $x+y+z=180°$, then $\cos 2x+\cos 2y-\cos 2z$ is equal to:

1. $4\sin x\sin y\sin z$

2. $1-4\sin x\sin y\cos z$

3. $4\sin x\sin y\sin z-1$

4. $\cos A\cos B\cos C$

Q. The angle at which curve $y=k{e}^{kx}$ intersects the y axis is

1. 
2. 
3. 
4. None of these

Q.

The angle of intersection between the curves $y=\left[\left|\sin x\right|+\left|\cos x\right|\right]$ and $x^2+y^2=10$, where$\left[x\right]$ denotes the greatest integer $\le x$ , is

1. ${\tan }^{-1}3$

2. ${\tan }^{-1}\left(\pm 3\right)$

3. ${\tan }^{-1}\left(\sqrt{3}\right)$

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

Q. The coordinates of the point P on the curve $y^2=2x^3$, at which the tangent is perpendicular to the line 4x - 3 y + 2 = 0, are given by

1. $(1,\sqrt{2})$
2. (2, 4)
3. $(\frac{1}{2},-\frac{1}{2})$
4. $(\frac{1}{8},-\frac{1}{16})$

Q.

If the area between $y=m{x}^2$ and $x=m{y}^2(m>0)$ is $\frac{1}{4}\mathrm{sq}$ units, then the value of $m$ is

1. $\pm 3\sqrt{2}$

2. $\pm \frac{2}{\sqrt{3}}$

3. $\sqrt{2}$

4. $\sqrt{3}$

Q.

The most general solutions of the equation $se{c}^{2}x=\sqrt{2}(1-{\tan }^{2}x)$ are given by

1. $n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{4}$

2. $2n\mathrm{\pi }+\frac{\mathrm{\pi }}{4}$

3. $n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{8}$

4. None of these

Q.

An equation of the tangent to curve $y=x^4$ from the point $(2,0)$on the curve is

1. $y=0$

2. $x=0$

3. $x+y=0$

4. None of these

Q.

The equation of the tangent to the curve $y=1-e^{x/2}$ at the point of intersection with the y- axis is

1. 2x +y =0

2. x - y =2

3. x +2y =0

4. None of these

Q. The shortest distance between the line $y=x$ and the curve $y^2=x-2$ is:

1. $\frac{11}{4\sqrt{2}}$
2. $\frac{7}{4\sqrt{2}}$
3. $\frac{7}{8}$
4. $2$

Q. Two forces $P$ and $2P$ are inclined at $120°$ with each other. If their resultant makes an angle $\alpha$ with their bisector then the value of $\alpha$ (in degree) is

1. $15°$
2. $30°$
3. $45°$
4. $60°$

Q.

The coordinates of the foot of the perpendicular from $(a,0)$ on the line $y=mx+\frac{a}{m}$ are

1. $\left(0,\frac{a}{m}\right)$

2. $\left(0,\frac{-a}{m}\right)$

3. $\left(\frac{a}{m},0\right)$

4. $\left(\frac{-a}{m},0\right)$

Q.

Find the coordinates of the points on the curve $y=x^2+3x+4$, the tangents at which pass through the origin.