Position of point wrt ellipse - class-XII
The dist.of a point P on the ellipse $\cfrac{{{x^2}}}{{12}} + \cfrac{{{y^2}}}{4} = 1$ from centre is $\sqrt 6 $ then the eccentric angle of P is
Point $(1,2)$ lies _____ the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$.
Eccentric angle of a point on the ellipse $x^{2}+3y^{2}=6$ at a distance $2$ units. from the centre of the ellipse is
Let the equation of the ellipse be $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. Let $f(x,y) = \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} - 1$. To determine whether the point $(x _1,y _1)$ lies inside the ellipse, the necessary condition is:
The locus of a point whose distance form the point $(3,0)$ is $3/5$ times its distance from the line $x=p$ is an ellipse with centre at the origin. The value of $p$ is
Point $\left(\sqrt5, \dfrac4{\sqrt5}\right)$ lies _____ the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{4} = 1$.
Determine position of a point $(2,3)$ with respect to the ellipse $\dfrac{x^{2}}{16}+\dfrac{y^{2}}{25}=1$.
Position of a point $(3,-4))$ with respect to the ellipse $16x^{2}+9y^{2}=144$ lies
The position of point $(4,3)$ with respect to the ellipse $\dfrac{x^2}{4}+\dfrac {y^2}{3}=1$
The distance of a point P on the ellipse $\dfrac{{{x^2}}}{{12}} + \dfrac{{{y^2}}}{4} = 1$ from centre is $\sqrt 6 $ then the ecentric angle of P is
Evaluate $\displaystyle \int x^2+3x+5\ dx$
An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is $\dfrac 23$ then the eccentricity of the ellipse is
The position of the point (1,3)n with respect to the ellipse $4x^{2}+9y^{2}-16x-54y+61=0$ is
If the point $(a\sin\theta, a\cos\theta)$ lies on the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ then the value of $\sin 2\theta$ is (where $a\neq b, a>0, b>0$ and $e$ is the eccentricity of the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$)
The locus of a point whose chord of contact to the ellipse $x^{2}+2y^{2}=1$ subtends a right angle at the centre of the ellipese is
Equation of the largest circle with centre (1,0) that can be inscribed in the ellipse $x^2 + 4y^2 = 16$ is
An ellipse of major axis $20\sqrt {3}$ and minor axis $20$ slides along the coordinate axes and always remains confined in the $1^{st}$ quadrant. The locus of the centre of the ellipse therefore describes the arc of a circle. The length of this arc is
A tangent to the ellipse $4x^2+9y^2=36$ is cut by tangent at the extremities of the major axis at $T$ and $T'$. The circles on $TT'$ as diameters passes through the point
If the line $x\, cos\, \alpha+y\,sin \,\alpha=p$ is normal to the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, then
Let $(a, 0)$ and $B(b, 0)$ be fixed distinct points on the $x-axis$, none of which coincides with the origin $O(0, 0)$ and let $C$ be a point on the $y-axis$. Let $L$ be a line through the $O(0, 0)$ and perpendicular to the line $AC$, The locus of the point of intersection of lines $L$ and $BC$ if $C$ varies along the $y-axis$, is (provided $x^{2}+ab\neq 0$)
If P($\theta$) and Q($\pi$/2 + $\theta$) are two points on the ellipse $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Locus of the mid-point of PQ is
The value of $\alpha$ for which the point $(\alpha,\alpha+2)$ is an interior point of smaller segment of the curve $x^{2}+y^{2}-4=0$ made by the chord of the curve whose equation is $3x+4y+12=0$ is
The distance of a point on the ellipse $\dfrac {x^{2}}{6}+\dfrac {y^{2}}{2}=1$ from the centre is $2$, then the eccentric angle is-
A rod of length $l$ rests against a vertical wall and a floor of a room.Let P be a point on the rod,nearer to its end on the wall, that divides its length in the ratio 1:2 if the rod begins to slide on the floor,then the locus of P is:
The distance from the foci of $P(a,b)$ on the ellipse $\dfrac {x^{2}}{9}+\dfrac {y^{2}}{25}=1$ are
The number of rational points on the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ is
In an ellipse the distance between its foci is 6 and its minor axis is 8 . Its eccentricity is
A point on the ellipse is $\displaystyle \frac{x^{2}}{6} + \frac{y^{2}}{2} = 1$ at a distance of $2$ from the centre of the ellipse has the eccentric angle
The position of the point $(1, 3)$ with respect to the ellipse $4x^2+9y^2-16x-54y+61=0$.
The point at shortest distance from the line x+y=7 and lying on an ellipse $x^2 + 2y^2 =6$, has coordinates
Which of the following points is an exterior point of the ellipse $\displaystyle 16 x^{2} + 9y^{2} - 16x - 32 = 0$.
An ellipse with foci $(0,\pm 2)$ has length of minor axis as $4$ units. Then the ellipse will pass through the point
Let a curve satisfying the differential equation $y^2dx+\left(x-\dfrac{1}{y}\right)dy=0$ which passes through $(1, 1)$. If the curve also passes through $(k, 2)$, then value of k is?
Let $E$ be the ellipse $\displaystyle \frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 4 } =1$ and $C$ be the circle ${ x }^{ 2 }+{ y }^{ 2 }=9$. Let $P$ and $Q$ be the points $(1,2)$ and $(2,1)$ respectively. Then
Find the equation of the ellipse whose eccentricity is $\dfrac{4}{5}$ and axes are along the coordinate axes and foci at $(0, \pm 4)$.
The point $(4, -3)$ with respect to the ellipse $4x^2+5y^2=1$.
Consider the ellipse with the equation $x^{2}+3y^{2}-2x-6y-2=0.$ The eccentric angle of a point on the ellipse at a distance 2 units from the contra of the ellipse is
Find the set of value(s) of $\alpha$ for which the point $\left ( 7\,-\, \displaystyle \frac{5}{4}\alpha,\,\alpha \right )$ lies inside the ellipse $\displaystyle \frac{x^2}{25}\,+\,\frac{y^2}{16}\,=\, 1.$
$\mathrm{A}$ssertion ($\mathrm{A}$): The point $(5,-2)$ lies outside the ellipse $24x^{2}+7y^{2}=12$.
Reason (R): lf the point $(x _{1},y _{1})$ lie outside the ellipse $\mathrm{S}=0$ then $S _{11}>0$
The point $(2\cos \theta , 3\sin \theta)$ lies ____________ the ellipse $\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$.
The distance of point '$\theta$' on the ellipse $\dfrac {x^2}{a^2} + \dfrac {y^2}{b^2}=1$ from a focus is:
$(2,3)$ lies _______ the ellipse $16 x^{2} + 9y^{2} - 16x - 32 = 0$
The point $(4\cos \theta , 4\sin \theta)$ lies ____________ the ellipse $\dfrac{x^2}{16}+\dfrac{y^2}{9}=1$
$(3,2)$ lies _______ the ellipse $16 x^{2} + 9y^{2} - 16x - 32 = 0$
The point $(1,1)$ lies ____________ the ellipse $\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$
The distance of a point $(\sqrt 6 \cos \theta, \sqrt 2 \sin \theta)$ on the ellipse $\dfrac {x^2}{6} + \dfrac {y^2}{2}=1$ from the centre is $2$, if:
Let $\dfrac {(x-3) ^2}9+\dfrac {(y-4) ^2}{16}=1$ then $(0,0)$ is
Let $5x^2+7y^2=140$, then $(3,-4)$ is:
Let $5x^2+7y^2=140$, then Position of $(4,-3)$ relative to the ellipse is
Let $\dfrac {(x-3) ^2}9+\dfrac {(y-4) ^2}{16}=1$ then $(3,4)$ is
Let $5x^2+7y^2=140$, then $(0,0)$ is:
Let $5x^2+7y^2=140$, then $(\sqrt {14},\sqrt {10})$ is:
If $P=(x, y), F _1=(3, 0), F _2=(-3, 0)$ and $ 16x^2+25y^2=400$, then $PF _1+PF _2$ equals
The position of the point $(1, 2)$ relative to the ellipse $2x^{2} + 7y^{2} = 20$ is
The minimum distance of origin from the curve $\frac{a^2}{x^2}+\frac{b^2}{y^2}=1$ is $(a>0,b>0)$
If $a$ and $c$ positive real number and the ellipse $\dfrac { { x }^{ 2 } }{ { 4c }^{ 2 } } +\dfrac { { y }^{ 2 } }{ { c }^{ 2 } } =1$ has four distinet points in common with the circle ${ x }^{ 2 }+{ y }^{ 2 }=9{ a }^{ 2 }$, then
An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is $2/3$ then the eccentricity of the ellipse is:
The segment of the tangent at the point P to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, intercepted by the auxiliary circle subtends a right angle at the origin. If the eccentricity of the ellipse is smallest possible, then the point P can be
If one end of the diameter of the ellipse $4x^2+y^2=16$ is $(\sqrt 3, 2)$, then the other end is:
The point P on the ellipse $4x^2+9y^2=36$ is such that the area of the $\Delta PF _1F _2=\sqrt{10} Sq$ units, where $F _1.F _2$ are Foci. Then P has the coordinates
Which of the following is an (x,y) coordinate pair located on the ellipse $4x^2 + 9y^2 = 100$?