Normal to an ellipse - class-XI
Description: normal to an ellipse | |
Number of Questions: 53 | |
Created by: Vaibhav Pathak | |
Tags: ellipse circles and conics section two dimensional analytical geometry-ii maths |
If a normal is drawn at point $P$ of ellipse $ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$, then the maximum distance from centre of ellipse will be $a-b$
If the normal at any point $P$ of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ meets the axes in $G$ and $g$ respectively, then $|PG| : |Pg|$ is equal to
One foot of normal of the ellipse $4x^2$ $+$ 9$y^2$ $= 36 $, that is parallel to the line $2x + y = 3 $, is
If the normal at any point on the ellipse $\dfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\dfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$ meets the axes in $G$ and $g$ respectively, then $PG:Pg=$
The equation of normal at the point $(0, 3)$ of the ellipse $9x^2 + 5y^2 = 45$ is
Find the equation of the normal to the ellipse $9x^2 + 16y^2 = 288$ at the point $(4, 3).$
The line $y=mx-\dfrac{\left(a^{2}-b^{2}\right)m}{\sqrt{a^{2}b^{2}m^{2}}}$ is normal to the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ for all values of $m$ belongs to
The number of normals to the ellipse $\dfrac { { x }^{ 2 } }{ 25 } +\dfrac { { y }^{ 2 } }{ 16 } =1$ which are tangents to the circle ${ x }^{ 2 }+{ y }^{ 2 }=9$ is
The equation of the normal to the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ at the end of latus rectum in quadrant $1^{st}$ and $4^{th}$ is
If line $y+3x=c$ is normal of the ellipse ${ x }^{ 2 }+3{ y }^{ 2 }=3$ then equation of normal is-
Length of latusrectum of the ellipse $\dfrac{x^{2}}{4}+\dfrac{y^{2}}{b^{2}}=1$, if the normal, at an end of latusrectum passes through one extremity of the minor axis, then equation of eccentricity of ellipse is
The normal of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at a point $P(x _1,y _1)$ on it, meets the x-axis in $G$. $PN$ is perpendicular to $OX$, where $O$ is origin. The value of $\frac{l(OG)}{l(ON)}$ is -
The maximum number of normals that can be drawn from any point outside of an ellipse, in general, is
The line $y = mx - \displaystyle \frac{(a^2 - b^2)m }{\sqrt{a^2+ b^2 m^2}}$ is normal to the ellipse $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ for all values of $m$ belongs to:
If the length of perpendicular drawn from origin to any normal to the ellipse $\cfrac{{x}^{2}}{16}+\cfrac{{y}^{2}}{25}=1$ is $l$, then $l$ cannot be
If the normal at an end of a latus-rectum of an ellipse $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through one extremity of the minor axis, the eccentricity of the ellipse is given by:
If the normal at one end of the latus rectum of an ellipse $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through one extremity of the minor axis, then:
The normal to the curve x$^2$ = 4y passing (1,2) is
The line $l x + m y = n$ is a normal to the ellipse $\dfrac { x ^ { 2 } } { a ^ { 2 } } + \dfrac { y ^ { 2 } } { b ^ { 2 } } = 1 ,$ if
The line $5x - 3y = 8\sqrt{2}$ is a normal to the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$. If $\theta$ be the eccentric angle of the foot of this normal , then '$\theta$' is equal to
Let $L$ be an end of the latus rectum of $y^2 = 4x$. The normal at $L$ meets the curve again at $M$. The normal at $M$ meets the curve again at $N$. The area of $\Delta LMN$ is
The line $2x+y =3$ cuts the ellipse $4x^2+y^2 =5$ at P and Q . If $\theta$ be the angle between the normals at these point then $tan \theta$ =
The equation of the normal to the ellipse $\displaystyle x^{2} + 4y^{2} = 16$ at the end of the latus rectum in the first quadrant is
If the tangent drawn at a point $\left( { t }^{ 2 },2t \right) $ on the parabola ${ y }^{ 2 }=4x$ is same as normal drawn at $\left( \sqrt { 5 } \cos { \alpha } ,2\sin { \alpha } \right) $ on the ellipse $\displaystyle \frac { { x }^{ 2 } }{ 5 } +\frac { { y }^{ 2 } }{ 4 } =1$, then which of following is true.
The number of distinct normal lines from the exterior point $\displaystyle \left ( 0, : c \right ), : c > b$ , to the ellipse $\displaystyle \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is
Equation of the normal to the ellipse $4 ( x - 1 ) ^ { 2 } + 9 ( y - 2 ) ^ { 2 } = 36 ,$ which is parallel to the line $3 x - y = 1 ,$ is
The number of normals that can be drawn to the curve $\displaystyle 4x^{2} + 9y^{2} = 36$ from an external point, in general, is
If the equation of normal to the ellipse $\displaystyle 4x^{2}+9y^{2}=36$ at the point $(3, -2)$ is $ px+qy=r$. Find the value of $p+q+r.$
Find the condition that the line $lx+my=n$ be a normal for ellipse
Find where the line $\displaystyle 2x+y=3$ cuts the curve $\displaystyle 4x^{2}+y^{2}=5.$ Obtain the equations of the normals at the points of intersection and determine the co-ordinates of the point where these normals cut each other.
If $y=mx+7\sqrt{3}$ is normal to $\dfrac{x^2}{18}-\dfrac{y^2}{24}=1$ then the value of m can be?
The normal at a point $P$ on the ellipse $x^{2}+4y^{2}=16$ meets the x-axis at $Q.$ If $M$ is the mid point of the line segment $PQ$, then locus of $M$ intersects the latus rectums of the given ellipse at the points.
The eccentric angle of the point where the line, $5x\, -\, 3y\, =\, 8\sqrt{2}$ is a normal to the ellipse $\displaystyle\frac{x^2}{25}\, +\, \frac{y^2}{9}\,=\,1$ is
On the ellipse $\displaystyle \frac { { x }^{ 2 } }{ 4 } +\frac { { y }^{ 2 } }{ 9 } =1$, one of the points at which the normals are parallel to the line $2x-y=1$ is
The equation of the normal to the ellipse $\displaystyle\frac{x^2}{a^2}\,+\,\frac{y^2}{b^2}\,=\,1$ at the positive end of latus rectum is :
Area of the triangle formed by the ${x}$ axis, the tangent and normal at $(3,2)$ to the ellipse $\displaystyle \frac{x^{2}}{18}+\frac{y^{2}}{8}=1$ is
Find the area of the rectangle formed by the perpendiculars from the center of the ellipse $\displaystyle \frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1$ to the tangent and normal at a point whose eccentric angle is $\displaystyle\frac{\pi}{4}.$
Assertion (A): Equation of the normal to the ellipse $\displaystyle \frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ at $P(\displaystyle \frac{\pi}{4})$ is $5x-3y-8\sqrt{2}=0$
Reason (R): Equation of the normal to the ellipse $\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at $P(x _{1},y _{1})$ is $\displaystyle \frac{a^{2}x}{x _1}-\frac{b^{2}y}{y _1}=a^{2}-b^2$
The maximum distance of any normal to the ellipse $\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ from the centre is:
The maximum distance of the normal to the ellipse $\displaystyle \frac{\mathrm{x}^{2}}{9}+\frac{\mathrm{y}^{2}}{4}=1$ from its centre is:
lf the tangent drawn at a point $(t^{2},2t)$ on the parabola $y^{2}=4x$ is same as normal drawn at $(\sqrt{5}\cos\alpha, 2\sin\alpha)$ on the ellipse $\displaystyle \frac{x^{2}}{5}+\frac{y^{2}}{4}=1$, then which of following is not true?
If the line $x\cos { \alpha } +y\sin { \alpha } =p$ be normal to the ellipse $\dfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\dfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$, then
If the line $x \cos a + y \sin a = p$ be normal to the ellipse $\dfrac{x^2}{a^2}$ $+\dfrac{y^2}{b^2}$ = 1 then
If the normal at the point $P(\theta)$ to the ellipse $\dfrac {x^{2}}{14} + \dfrac {y^{2}}{5} = 1$ intersects it again at the point $Q(2\theta)$, then $\cos \theta$ is equal to
The number of tangents to the circle ${x}^{2}+{y}^{2}=3$ that are normals to the ellipse $\cfrac{{x}^{2}}{9}+\cfrac{{y}^{2}}{4}$ is
Which of the following is/are true?
Number of distinct normal lines that can be drawn to the ellipse $\displaystyle \frac{x^2}{169} + \frac{y^2}{25} = 1$ from the point $P(0, 6)$ is:
If the normal at any point $P$ on the ellipse $\displaystyle\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ meets the axes in $G$ and $g$ respectively, then $PG:Pg=$
The eccentricity of an ellipse whose centre is at the origin is $\dfrac{1}{2}.$ If one of its directrices is $x = - 4,$ then the equation of the normal to it at $\left( {1,\dfrac{3}{2}} \right)$ is
Tangents are drawn to the ellipse $ \displaystyle \frac{x^2}{a^2}+\displaystyle \frac{y^2}{b^2}=1 $ at points where it is intersected by the line $ \ell x+my+n=0 $. Find the point of intersection of tangents at these points.
A ray emanating from the point $(4, 0)$ is incident on the ellipse $9x^2\, +\, 25y^2\, =\, 225$ at the point $P$ with abscissa $3$. Find the equation of the reflected ray after first reflection.
The tangent and normal to the ellipse $x^2\, +\, 4y^2\, =\, 4$ at a point $P(\theta)$ on it meet the major axis in $Q$ and $R$ respectively. If $QR = 2$, the eccentric angle $\theta$ of $P$ is given by