Chord of Contact: Ellipse

Q.

The length of minor axis (along $y$-axis) of an ellipse of the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line $x+6y=8$, then its eccentricity is

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

2. $\frac{1}{2}\sqrt{\frac{11}{3}}$

3. $\frac{1}{3}\sqrt{\frac{11}{3}}$

4. $\sqrt{\frac{5}{6}}$

Q.

If a tangent to the curve$y=6x–x^2$ is parallel to the line$4x–2y–1=0$, then the point of tangency on the curve is

1. $(2,8)$

2. $(8,2)$

3. $(6,1)$

4. $(4,2)$

Q.

If a tangent to the curve$y=2+\sqrt{4x+1}$ has slope $2/5$ at a point, then the point is

1. $(0,2)$

2. $(3/4,4)$

3. $(2,5)$

4. $(7,6)$

5. $(6,7)$

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. 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

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

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

Q. If the equation of the chord of contact of tangents drawn from a point $(a,b)$ to an ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ is given by $3x+4y=6.$ Then, find the point $(a,b).$

1. $(8,6)$
2. $(5,6)$
3. $(2,5)$
4. $(4,3)$

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

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

Q.

If $m=\tan \left(x\right)$is the slope of the tangent to the curve $e^y=1+x^2$ then

1. $\left|\tan \left(x\right)\right|>1$

2. $\left|\tan \left(x\right)\right|<1$

3. $-1\le \tan \left(x\right)\le 1$

4. $-1\le \left|\tan \left(x\right)\right|\le 1$

Q. Equation of chord of contact of tangents drawn from a point $(m,n)$ to hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ is given by $3x-4y=6$. Then evaluate the coordinates of $(m,n)$.

1. $P(6,8)$
2. $P(8,6)$
3. $P(4,3)$
4. $P(3,4)$

Q.

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

1. $x^2-y^2=9$

2. $x^2+y^2=41$

3. $x^2+y^2=9$

4. $x^2+y^2=25$

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. 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. $2p-4=h$