Normal to a hyperabola- class-XII
Description: normal to a hyperabola | |
Number of Questions: 35 | |
Created by: Avani Handa | |
Tags: hyperbola two dimensional analytical geometry-ii maths |
The equation of the curve which is such that the protion of the axis of x cut off between the origin and tangent at any point is proportional to the ordinate of that point is _______________.
The hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, normals are drawn to curve $\left( {{{\left( {\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}} \right)}^2} - 1} \right)\left( {\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}} \right) = 0$.
Find the sum; of abscissa of foot of all such normals.
If the straight line $(a - 2) x - by + 4 = 0$ is normal to the hyperbola $xy = 1$ then which of the followings does not hold?
The normal to the hyperbola $4x^2-9y^2=36$ meets the axes in $M$ and $N$ and the lines $MP$, $NP$ are drawn right angles at the axes. The locus of $P$ is the hyperbola
A normal to the hyperbola, $4x^2-9y^2=36$ meets the co-ordinate axes x and y at A and B, respectively. If the parallelogram $OABP$($O$ being the origin) is formed, then the locus of $P$ is?
Equation of the normal to the hyperbola $3x^2-y^2=3$ at $(2, -3)$ is?
Line x cos$\alpha $+yin$\alpha $=p is a normal to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $, if
Line $ x \cos \alpha + y \sin \alpha = p $ is a normal to the hyperbola $ \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 $, if
A straight line is drawn parallel to the conjugate axis of the hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$ to meet it and the conjugate hyperbola respectively in the point $P$ and $Q$. The normals at $p$ and $Q$ to the curves meet on
If the normal at $\left (ct _1,\dfrac { c}{t _1}\right)$ on the hyperbola $xy = c^2$ cuts the hyperbola again at $\left (ct _2, \dfrac {c}{t _2}\right)$, then $t _2^3 t _2$ $=$
If the tangent and normal to a rectangular hyperbola cut off intercepts $x _1$ and $x _2$ on one axis and $y _1$ and $y _2$ on the other axis, then
The number of normal to the hyperbola $\cfrac { { x }^{ 2 } }{ { a }^{ 2 } } -\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$ from an external point is
The normal to the rectangular hyperbola $xy=-c^2$ at the point $'t _1'$ meets the curve again at the point $'t _2'$. The value of $t _1^3 \cdot t _2$ is
If the normal at '$\theta $' on the hyperbola $\cfrac { { x }^{ 2 } }{ { a }^{ 2 } } -\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$ meets the transverse axis at $G$ and $A$ and $A'$ are the vertices of the hyperbola, then $AG.A'G$ $=$
If the normal at $'\theta'$ on the hyperbola $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ meets the transverse axis at G, and A and A' are the vertices of the hyperbola, then AG.A'G $=$
The equation of normal at $\left( at,\dfrac { a }{ t } \right)$ to the hyperbola $xy={ a }^{ 2 }$ is ________________________.
The normal at P to a hyperbola of eccentricity e, intersects its transverse and conjugate axes at L and M respectively. If locus of the mid-point of LM is a hyperbola, then eccentricity of the hyperbola is
The maximum number of normals to the hyperbola $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2}=1$ from an external point is :
Set of value of h for which the number of distinct common normals of $(x-2)^{ 2 }=4 (y-3)$ and ${ x }^{ 2 }+{ y }^{ 2 }-2x-hy-c=0$ where, $\left( c>0 \right) $ is 3, is
The length of sub normal to the curve $xy={ a }^{ 2 }$ at (x,y) on it varies at
Let $P (a\sec \theta , b\tan \theta ) $ and $Q\left ( a\sec \phi , b\tan \phi \right )$ where $\theta +\phi =\pi /2$, be two points on the hyperbola $x^{2}/a _{2}-y _{2}/b _{2}=1$. If (h, k) is the point of intersection of normals at P and Q, then k is equal to
Find the equation of normal to the hyperbola $\displaystyle \frac{x^2}{25}\, -\, \displaystyle \frac{y^2}{16}\, =\, 1$ at $(5, 0)$.
Find the equation of normal to the hyperbola $\displaystyle \frac{x^2}{16}\, -\displaystyle
\frac{y^2}{9}=1$ at the point $\left ( 6, \displaystyle \frac{3}{2}\sqrt{5}\,\right )$
If e and e' be the eccentricities of a hyperbola and its conjugate, then $\displaystyle \dfrac{1}{e^2} + \dfrac{1}{e'^2} $ is equal to
The normal to a curve at $P(x, y)$ meets the x-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is :
lf the line $ax+by+c=0$ is a normal to the curve $xy=1$, then :
The equation of the normal at the positive end of the latusrectum of the hyperbola $x^2-3y^2=144$ is
Which one of the following points does not lie on the normal to the hyperbola, $\cfrac { { x }^{ 2 } }{ 16 } -\cfrac { { y }^{ 2 } }{ 9 } =1$ drawn at the point $\left( 8,3\sqrt { 3 } \right) $?
Let $A\left( A\sec { \theta } ,3\tan { \theta } \right) $ and $B\left( A\sec { \phi } ,3\tan { \phi } \right) $ where $\theta +\phi =\cfrac { \pi }{ 2 } $, be two points on the hyperbola $\cfrac { { x }^{ 2 } }{ 4 } -\cfrac { { y }^{ 2 } }{ 9 } =1$. If $\left( \alpha ,\beta \right) $ is the point of intersection of normals to the hyperbola at $A$ and $B$, then $\beta=$
If the sum of the slopes of the normal from a point P to the hyperbola $xy = {c^2}$is equal to $\lambda (\lambda \in {R^ + })$,then the locus of point P is
Let $P\left( a\sec { \theta } ,b\tan { \theta } \right) $ and $Q\left( a\sec { \phi } ,b\tan { \phi } \right) $, where $\theta +\phi =\dfrac {\pi}{2} $, be the two points on the hyperbola $\cfrac { { x }^{ 2 } }{ { a }^{ 2 } } -\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$. If $(h,k)$ is the point of intersection of the normals of $P$ and $Q$, then $k$ is equal to
If a normal of slope $m$ to the parabola ${ y }^{ 2 }=4ax$ touches the hyperbola ${ x }^{ 2 }-{ y }^{ 2 }={ a^2 }$, then
If a normal of slope $m$ to the parabola $y^2 = 4ax$ touches the hyperbola $x^2 - y^2 = a^2$, then
Let P $(asec \theta,\, btan \theta)$ and Q $(asec \phi,\, btan \phi)$, where $\theta\, +\, \phi\, =\, \displaystyle \frac{\pi}{2}$, be two points on the hyperbola $\displaystyle \frac{x^2}{a^2}\, -\, \frac{y^2}{b^2}\, =\, 1$. If (h, k) is the point of intersection of the normals at P & Q, then k is equal to
From any point R two normals which are right angled to one another are drawn to the hyperbola $\displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1,\left ( a>b \right )$ If the feet of the normals are P and Q then the locus of the circumcentre of the triangle PQR is