Point Form of Normal:Hyperbola

Q. The equation of the normal to the curve $\frac{x^2}{16}-\frac{y^2}{9}=1$ at $(8,3\sqrt{3})$ is

1. $4x-\sqrt{3}y=23$
2. $2x+\sqrt{3}y=25$
3. $3x-2\sqrt{3}y=6$
4. $x+\sqrt{3}y=17$

Q. Find the slope of the line 3 x-4 Y - 10 is equal to zero?

Q.

The pair of straight lines joining the origin to the points of intersection of the line $y=2\sqrt{2}x+c$ and the circle $x^2+y^2=2$ are at right angles, if

1. $c^2-4$

2. $c^2-8$

3. $c^2-9$

4. $c^2-10$

Q.

Let $P(3,3)$ be a point on the hyperbola, $\left(\frac{x^2}{a^2}\right)-\left(\frac{y^2}{b^2}\right)=1$. If the normal to it at $P$ intersects the x-axis at $(9,0)$ and $e$ is its eccentricity, then the ordered pair $(a^2,e^2)$ is equal to

1. $\left(9,3\right)$

2. $\left(\frac{9}{2},2\right)$

3. $\left(\frac{9}{2},3\right)$

4. $\left(\frac{3}{2},2\right)$

Q.

Let P(6, 3) be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

If the normal at the point P intersects the x-axis at (9, 0), then the eccentricity of hyperbola is,

1. 

2. 

3. 

4. 

Q. The locus of the middle points of chords of hyperbola $3x^2-2y^2+4x-6y=0$ parallel to $y=2x$ is :

1. $3x-4y=4$
2. $3x+4y=8$
3. $5x-4y=9$
4. $6x-3y=4$

Q. Let f(x) is continuous function as shown in figure. If the area bounded by the curve $y=f\left(x\right)$, $y=x\sqrt{x}$ and line segment AB is equal to area bounded by $y=f\left(x\right)$, y-axis and line segment AC and ${\int }_0^{1}f\left(x\right)dx=\frac{1}{3}$ and $f\left(\frac{1}{4}\right)=\frac{a}{b}$ (where a and b have no common factors) then value of $\lfloor a/b\rfloor$ is (where $\lfloor \rfloor$ denotes greatest integer function)

Q.

The distance between the origin and the normal to the curve $y=e^{2x}+x^2$at $x=0$ is

1. $2$

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

3. $\frac{2}{\sqrt{5}}$

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

5. $\frac{1}{\sqrt{5}}$

Q. If a normal drawn at one end of the latus rectum of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the axes at points $A\&B$ respectively, then area of $△OAB$ (in sq.units) is

1. $a^2{e}^5$
2. $\frac{a^2{e}^5}{2}$
3. $\frac{a^2{e}^5}{4}$
4. $\frac{a^2{e}^5}{8}$

Q.

The equation of the normal at the point (6, 4) on the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=3$ is

1. 3x - 8y = 50

2. 3x + 8y = 50

3. 8x - 3y = 50

4. 8x + 3y = 50

Q. The range of parameter ${}^{\prime }{a}^{\prime }$ for which a unique circle will pass through the points of intersection of the rectangular hyperbola $x^2-y^2=a^2$ and the parabola $y=2x^2$, is

1. $a\in R$
2. $a\in (-1,1)$
3. $a\in (-\frac{1}{2},\frac{1}{2})$
4. $a\in (-\frac{1}{4},\frac{1}{4})$

Q. The equation of the normal to the hyperbola $x^2-9y^2=7$ at the point (4, 1) is .

1. 9x + 4y = 40
2. 9x - 4y = 40
3. 4x + 9y = 40
4. 4x - 9y = 40

Q.

Let P(6, 3) be a point on hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$
If the normal at the point P intersect the x - axis at (9, 0), then the eccentricity of the hyperbola is

1. $\sqrt{\frac{5}{2}}$

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

3. $\sqrt{2}$

4. $\sqrt{3}$

Q. If $a^2+\frac{1}{a^2}=11$, then find the value of $(a-\frac{1}{a})$

Q.

The equation of the normal to the curve $y^4=a{x}^3$ at $(a,a)$ is

1. $x+2y=3a$

2. $3x–4y+a=0$

3. $4x+3y=7a$

4. $4x–3y=0$

Q. The equation of the normal to the curve $\frac{x^2}{16}-\frac{y^2}{9}=1$ at $(8,3\sqrt{3})$ is

1. $4x-\sqrt{3}y=23$
2. $2x+\sqrt{3}y=25$
3. $3x-2\sqrt{3}y=6$
4. $x+\sqrt{3}y=17$

Q.

How do you find $a$ and $b$ of a Hyperbola?

Q.

How do you find the normal line of a function?

Q. Let f be a real valued function satisfying $f\left(\frac{x}{y}\right)=f\left(x\right)-f\left(y\right)$ and $\underset{x\to 0}{lt}\frac{f(1+x)}{x}=3$. Then find the area bounded by the curve y=f(x), the y-axis and the line y=3 in sq units

1. $3^e$
2. $e^3$
3. $3e$
4. $e+3$

Q. Let $P(8,4)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal at point $P$ intersects the $x-$axis at $(12,0)$, then the value of eccentricity is

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

Q.

What is the equation of a normal line?

Q. Find the eccentricity, coordinates of foci, equation of directrix and length of the latus-rectum of the hyperbola $2x^2-3y^2=5$

Q.

If the normal to the curve $y=f\left(x\right)$ at the point $(3,4)$ makes $\frac{3\mathrm{\pi }}{4}$ with the positive $x-$axis then $f\left(3\right)$ $=?$

1. $-1$

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

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

4. $1$

Q.

The normal at the point (1, 1) on the curve 2y + x² = 3 is

(A) x + y = 0 (B) x − y = 0

(C) x + y + 1 = 0 (D) x − y = 1

Q. The slope of the tangent to the curve at any point is the reciprocal of twice the ordinate at that point. The curve passes through the point (4, 3). Determine its equation.

Q.

The point on the curve $3y=6x-5x^3$, the normal at which passes through the origin is,

1. $\left(1,\frac{1}{3}\right)$

2. $\left(\frac{1}{3},1\right)$

3. $\left(2,-\frac{28}{3}\right)$

4. None of these.

Q. The eccentricity of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is:

1. $\frac{\sqrt{5}}{2}$
2. $\frac{2}{3}$
3. $\frac{4}{9}$
4. $\frac{\sqrt{5}}{3}$

Q.

The equation of the normal to the curve $y^4=a{x}^3$ at $\left(a,a\right)$ is

1. $x+2y=3a$

2. $3x-4y+a=0$

3. $4x+3y=7a$

4. $4x-3y=0$

Q.

Find the equation of the hyperbola satisfying the give conditions: Foci, passing through (2, 3)

Q. Through a point P(f, g, h) a plane is drawn at right angles to OP, to meet the axes in A, B, C. If OP = r, the
centroid of the triangle ABC is

1. $(\frac{f}{3r},\frac{g}{3r},\frac{h}{3r})$
2. $(\frac{r^2}{3f^2},\frac{r^2}{3g^2},\frac{r^2}{3h^2})$
3. $(\frac{r^2}{3f},\frac{r^2}{3g},\frac{r^2}{3h})$
4. $(\frac{r^2}{f},\frac{r^2}{g},\frac{r^2}{h})$