Slope Form of Tangent: Ellipse

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 equation of a tangent parallel to y = x drawn to $\frac{x^2}{3}-\frac{y^2}{2}=1$, is

1. x -y + 1 = 0
2. x - y + 2 = 0
3. x + y - 1 = 0
4. x + y – 2 = 0

Q.

The minimum area of triangle formed by the tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and coordinate axes is:

1. ab sq. units

2. $\frac{a^2+ab+b^2}{3}$

3. $\frac{(a+b{)}^2}{2}sq.units$

4. $\frac{a^2+b^2}{2}sq.units$

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. If a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ having slope $2$ is normal to the circle $x^2+y^2+4x+1=0,$ then the maximum value of $ab$ is

Q. If the normal to the ellipse $3x^2+4y^2=12$ at a point $P$ on it is parallel to the line, $2x+y=4$ and the tangent to the ellipse at $P$ passes through $Q(4,4)$ then $PQ$ is equal to

1. $\frac{5\sqrt{5}}{2}$ unit
2. $\frac{\sqrt{157}}{2}$ unit
3. $\frac{\sqrt{61}}{2}$ unit
4. $\frac{\sqrt{221}}{2}$ unit

Q. An ellipse has OB as semi-minor axis, F and F’ its foci and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is

1. 
2. 
3. 
4. 

Q.

The minimum area of triangle formed by the tangents to the ellipse
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and coordinate axes is:

1. $absq.units$

2. $\frac{(a+b{)}^2}{2}sq.units$

3. $\frac{a^2+b^2}{2}sq.units$

4. $\frac{a^2+ab+b^2}{3}$

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.

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

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