Chord of Contact: Ellipse

Q. If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint $(h,k)$, then which of the following is(are) possible value(s) of $p,h\text{and}k$ ?

1. $p=-2,h=2,k=-4$
2. $p=-1,h=1,k=-3$
3. $p=2,h=3,k=-4$
4. $p=5,h=4,k=-3$

Q. If from a point $P,$ tangents $PQ$ and $PR$ are drawn to the ellipse $\frac{x^2}{2}+y^2=1,$ such that equation of $QR$ is $x+3y=1,$ then the coordinates of $P$ is

1. $(1,-2)$
2. $(2,3)$
3. $(2,-3)$
4. $(1,2)$

Q. If the chord through the points whose eccentric angles are $\theta$ and $\varphi$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ passes through a focus, then the value of $\tan \left(\frac{\theta }{2}\right)\tan \left(\frac{\varphi }{2}\right)$is

1. $\frac{1}{9}$
2. $-9$
3. $-\frac{1}{9}$
4. $9$

Q. Locus of the point of intersection of the tangents at the end points of a focal chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is/are

1. auxiliary circle of the ellipse
2. ellipse
3. director dircle of ellipse
4. directrices of ellipse

Q. The locus of the centre of the circles which touch both the circles $x^2+y^2=a^2$ and $x^2+y^2=4ax$ externally is

1. a circle
2. a parabola
3. an ellipse
4. a hyperbola

Q. If Tangents $PA$ and $PB$ are drawn to ellipse
$\frac{x^2}{16}+\frac{y^2}{9}=1$ from a point $P(0,5),$ then area of triangle $PAB$ is equal to

1. $\frac{256}{25}$ sq. unit
2. $\frac{32}{5}$ sq. unit
3. $\frac{1024}{25}$ sq. unit
4. $\frac{16}{5}$ sq. unit

Q. Form point $P(8,27)$, tangent $PQ$ and $PR$ are drawn to the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$.If the angle subtended by $QR$ at origin is $\varphi$, then $\tan \varphi =$

1. $\frac{\sqrt{6}}{65}$
2. $\frac{4\sqrt{6}}{65}$
3. $\frac{8\sqrt{6}}{65}$
4. $\frac{48\sqrt{6}}{455}$

Q. Tangents are drawn to the ellipse $\frac{x^2}{36}+\frac{y^2}{9}=1$ from any point on the parabola $y^2=4x.$ The corresponding chord of contact will touch a parabola, whose equation is

1. $y^2+4x=0$
2. $y^2-4x=0$
3. $4y^2+9x=0$
4. $y^2+9x=0$

Q. Equation of the chord of contact, drawn to the ellipse $4x^2+9y^2=36$ from the point $(m,n)$ where $m\cdot n=m+n$ and $m,n\in I^{+}$ is

1. $4x+9y=9$
2. $2x+2y=1$
3. $4x+9y=18$
4. $4x+9y=36$

Q.

The locus of the point of intersection of the tangents at the extremities of the chord of the ellipse $x^2+2y^2=6$ which also touches the ellipse $x^2+4y^2=4,$ is

1. $x^2+y^2=4$
2. $x^2+y^2=6$
3. $x^2+y^2=9$
4. $x^2+y^2=12$

Q. Let $P$ be any point on any directrix of an ellipse. Then chords of contact of point $P$ with respect to the ellipse and its auxiliary circle intersect at

1. some point on the major axis depending upon the position of point $P$
2. midpoint of the line segment joining the centre to the corresponding focus
3. corresponding focus
4. midpoint of the line segment joining the directrix to the corresponding focus

Q. Chord of contact is drawn from point $P$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ which forms a triangle of constant area with the coordinate axes. Then the locus of point $P$ is

1. $xy=c^2$
2. $x+y=c^2$
3. $x^2+y^2=c^2$
4. $x+\frac{1}{y}=c^2$

Q. If tangent to parabola $y^2=4ax$ intersects ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at $A$ and $B,$ then the locus of point of intersection of tangents at $A$ and $B,$ is

1. straight line
2. ellipse
3. parabola with length of latus rectum $\frac{a^4}{b^3}$ unit
4. parabola with length of latus rectum $\frac{b^4}{a^3}$ unit

Q. If $2x+y=p$ is a chord to the parabola $y^2=16x$ whose midpoint is $(h,k)$, then which of the following is/are true?

1. $k=4$
2. $k=-4$
3. $2h-4=p$
4. $2p-4=h$

Q. From any point on the line $y=x+4$, tangents are drawn to the auxiliary circle of the ellipse $x^2+4y^2=4.$ If $P,Q$ are the points of contact and $A,B$ are the corresponding points of $P$ and $Q$ on the ellipse respectively, then the locus of the midpoint of $AB$ is

1. $4x^2+y^2+y-2x=0$
2. $4y^2+x^2+x-2y=0$
3. $4x^2+y^2+y+2x=0$
4. $4y^2+x^2+x+2y=0$

Q.

Find the maximum area of an isosceles triangle inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with its vertex at one end of the major axis.

Q. The ellipse $E_1:\frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point $(0,4)$ circumscribes the rectangle $R.$ The Eccentricity of the ellipse $E_2$ is

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

Q. From a point, perpendicular tangents are drawn to the ellipse $x^2+2y^2=2.$ The chord of contact touches a circle concentric with the given ellipse. The ratio of the maximum, minimum areas of the circle is

Q. Let chord $AB$ of the parabola $y^2=4x$ subtend ${90}^{\circ }$ at the vertex. If the coordinates of $A$ and $B$ are $(t_1^2,2t_1)$ and $(t_2^2,2t_2)$ respectively, then

1. $t_1{t}_2=-1$
2. $t_1{t}_2=-2$
3. $t_1{t}_2=-4$
4. $t_1{t}_2=-8$

Q. Let $P$ be any point on any directrix of an ellipse. Then chords of contact of point $P$ with respect to the ellipse and its auxiliary circle intersect at

1. some point on the major axis depending upon the position of point $P$
2. midpoint of the line segment joining the centre to the corresponding focus
3. corresponding focus
4. midpoint of the line segment joining the directrix to the corresponding focus

Q. The equation of the chord joining two points$(x_1,y_1)and(x_1,y_1)ontherectangularhyperbolaxy=c^{2}is$

1. 
2. 
3. 
4. 

Q.

The locus of the point of intersection of the tangents at the extremities of the chord of the ellipse $x^2+2y^2=6$ which touches the ellipse $x^2+4y^2=4,$ is

1. $x^2+y^2=4$
2. $x^2+y^2=6$
3. $x^2+y^2=9$
4. $x^2+y^2=12$

Q.

The Equation of chord of contact of $\frac{x^2}{25}+\frac{y^2}{16}=1is4x+5y-20=0$ Find the point from which the tangents are drawn

1. (4, 6)

2. (5, 4)

3. (4, 5)

4. (6, 5)

Q. The locus of point of intersection of tangents to the circle $x=a\cos \theta ,y=a\sin \theta$ at points whose parametric angles differ by $\pi /4$ is?

1. $x^2+y^2=2(2-\sqrt{2})a^2$
2. $x^2+y^2=2(\sqrt{2}-1{)}^2{a}^2$
3. $x^2+y^2=(\sqrt{2}+1{)}^2{a}^2$
4. $9(x^2+y^2)=4a^2$

Q.

The straight lines joining the origin to the points of intersection of the line 2x + y = 1 and curve $3x^2+4xy-4x+1=0$ include an angle

1. 

2. 

3. 

4. 

Q. Let $\Gamma$ be a circle with diameter $AB$ and centre $O$. Let $l$ be the tangent to $\Gamma$ at $B$. For each point $M$ on $\Gamma$ different from $A$, consider the tangent $t$ at $M$ and let it intersect $l$ at $P$. Draw a line parallel to $AB$ through $P$ intersecting $OM$ at $Q$. The locus of $Q$ as $M$ varies over $\Gamma$ is

1. an arc of a circle
2. a parabola
3. an arc of an ellipse
4. a branch of a hyperbola

Q.

What is the equation of chord of contact of tangents drawn from P(10, 8) to the ellipse
$\frac{x^2}{25}+\frac{y^2}{16}=1$.

1. $4x+5y=10$

2. $5x-4y=10$

3. $4x-5y=5$

4. $4x-5y=10$

Q. Form point $P(8,27)$, tangent $PQ$ and $PR$ are drawn to the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$.If the angle subtended by $QR$ at origin is $\varphi$, then $\tan \varphi =$

1. $\frac{\sqrt{6}}{65}$
2. $\frac{4\sqrt{6}}{65}$
3. $\frac{8\sqrt{6}}{65}$
4. $\frac{48\sqrt{6}}{455}$

Q.

The locus of the point of intersection of the tangents at the extremities of the chord of the ellipse $x^2+2y^2=6$ which touches the ellipse $x^2+4y^2=4,$ is

1. $x^2+y^2=4$
2. $x^2+y^2=6$
3. $x^2+y^2=9$
4. $x^2+y^2=12$

Q.

Which of the following gives the equation of director circle of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$?

1. x² - y² = 9

2. x² + y² = 41

3. x² + y² = 9

4. x² + y² = 25