Slope Form of Tangent: Ellipse

Q.

If the line $ax+by+c=0$ is tangent to the curve $xy=4$, then

1. $a<0,b>0$

2. $a=0,b>0$

3. $a<0,b<0$

4. $a=0,b<0$

Q.

The equation of the tangent of the curve $y=x+\frac{4}{x^2}$ that is parallel to the x-axis is

1. $y=0$

2. $y=1$

3. $y=2$

4. $y=3$

Q. If $3x+4y=12\sqrt{2}$ is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{9}=1$ for some $a\in R$, then the distance between the foci of the ellipse is :

1. $2\sqrt{5}$
2. $2\sqrt{7}$
3. $2\sqrt{2}$
4. $4$

Q. Let $E$ be the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and $C$ be the circle $x^2+y^2=9$. Let $P$ and $Q$ be the points $(1,2)$ and $(2,1)$ respectively. Then

1. $Q$ lies inside $C$ but outside $E$
2. $Q$ lies outside both $C$ and $E$
3. $P$ lies inside both $C$ and $E$
4. $P$ lies inside $C$ but outside $E$

Q. A straight line $PQ$ touches the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and the circle $x^2+y^2=r^2(3<r<4).$ $RS$ is chord of the circle which is parallel to $PQ$ and passes through any focus of ellipse, then the length of $RS$ is equal to unit.

Q. The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2+3y^2=6$ on any tangent to it is:

1. $(x^2-y^2{)}^2=6x^2+2y^2$
2. $(x^2-y^2{)}^2=6x^2-2y^2$
3. $(x^2+y^2{)}^2=6x^2+2y^2$
4. $(x^2+y^2{)}^2=6x^2-2y^2$

Q. Given ellipse $x^2+4y^2=16$ and parabola $y^2-4x-4=0.$
The quadratic equation whose roots are the slopes of the common tangents to the parabola and the ellipse, is

1. $3x^2-1=0$
2. $5x^2-1=0$
3. $15x^2+2x-1=0$
4. $2x^2-1=0$

Q. The equation of the ellipse, whose axes are coincident with the co-ordinates axis and which touches the straight lines $3x-2y-20=0$ and $x+6y-20=0,$ is

1. $\frac{x^2}{40}+\frac{y^2}{10}=1$
2. $\frac{x^2}{5}+\frac{y^2}{8}=1$
3. $\frac{x^2}{10}+\frac{y^2}{40}=1$
4. $\frac{x^2}{40}+\frac{y^2}{30}=1$

Q. If the tangents on the ellipse $4x^2+y^2=8$ at the point $(1,2)$ and $(a,b)$ are perpendicular to each other, then $a^2$ is equal to:

1. $\frac{2}{17}$
2. $\frac{128}{17}$
3. $\frac{4}{17}$
4. $\frac{64}{17}$

Q. Tangents are drawn to the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ at the end points of the latus rectum. The area of quadrilateral formed by these tangents is

1. $\frac{64}{3}$ sq. units
2. $\frac{128}{3}$ sq. units
3. $\frac{256}{3}$ sq. units
4. $\frac{32}{3}$ sq. units

Q. If the ellipse $\frac{x^2}{a^2-3}+\frac{y^2}{a+4}=1$ is inscribed in a square of side length $a\sqrt{2}$ then a is

1. 4
2. 2
3. 1
4. None of these

Q. The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2+3y^2=6$ on any tangent to it is:

1. $(x^2-y^2{)}^2=6x^2+2y^2$
2. $(x^2-y^2{)}^2=6x^2-2y^2$
3. $(x^2+y^2{)}^2=6x^2+2y^2$
4. $(x^2+y^2{)}^2=6x^2-2y^2$

Q. A tangent having slope $-2$ to the ellipse $\frac{x^2}{18}+\frac{y^2}{36}=1$ intersects the major axis and minor axis at $A$ and $B$, respectively. If $C(0,0)$ is the centre of the ellipse, then the area of $△ABC$ is

1. $12$ sq. units
2. $24$ sq. units
3. $36$ sq. units
4. $27$ sq. units

Q. Number of real tangents, which can be drawn from the point $(4,3)$ to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1,$ is

Q. A series of concentric ellipses $E_1,E_2,\dots ,E_n$ are drawn such that $E_n$ touches the extremities of the major axis of $E_{n-1}$ and the foci of $E_n$ coincide with the extemities of minor axis of $E_{n-1}.$ If the eccentricites of the ellipse are independent of $n,$ then the value of the eccentricity, is

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

Q. If $3x+4y=12\sqrt{2}$ is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{9}=1$ for some $a\in R$, then the distance between the foci of the ellipse is :

1. $2\sqrt{5}$
2. $2\sqrt{7}$
3. $2\sqrt{2}$
4. $4$

Q. Let $F_1(x_1,0)$ and $F_2(x_2,0),$ for $x_1<0$ and $x_2>0,$ be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1.$ Suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.

If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to tha parabola at $M$ meets the $x-$axis at $Q$, then the ratio of area of the triangle $MQR$ to area of the quadrilateral $M{F}_{1}N{F}_2$ is

1. $3:4$
2. $4:5$
3. $5:8$
4. $2:3$

Q.

The equation of tangent with slope $m$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is given by $y=mx\mp \sqrt{a^2+b^2{m}^2}.$

1. True

2. False