Equation of Pair of Tangents: Ellipse

Q. An angle of intersection of the curves, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $x^2+y^2=ab,$ $a>b$, is

1. ${\tan }^{-1}\left(\frac{a+b}{\sqrt{ab}}\right)$
2. ${\tan }^{-1}\left(\frac{a-b}{2\sqrt{ab}}\right)$
3. ${\tan }^{-1}\left(\frac{a-b}{\sqrt{ab}}\right)$
4. ${\tan }^{-1}\left(2\sqrt{ab}\right)$

Q. The locus of point of intersection of pair of tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ if the sum of ordinates of their point of contact is half the length of minor axis, is

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{2x}{a}=0$
2. $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{2x}{a}=0$
3. $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{2y}{b}=0$
4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{2y}{b}=0$

Q. The equation of tangent to the curve $y=2cosx$ at $x=\frac{\pi }{4}$ is

1. $y-\sqrt{2}=2\sqrt{2}(x-\frac{\pi }{4})$
2. $y+\sqrt{2}=\sqrt{2}(x+\frac{\pi }{4})$
3. $y-\sqrt{2}=-\sqrt{2}(x-\frac{\pi }{4})$
4. $y-\sqrt{2}=\sqrt{2}(x-\frac{\pi }{4})$

Q. Find the common tangent to the ellipses $\frac{x^2}{16}+\frac{y^2}{9}=1$ and $\frac{x^2}{9}+\frac{y^2}{16}=1$.

1. $x+y+5=0$
2. $x-y+5=0$
3. $x-y-5=0$
4. All of the above.

Q. Tangents are drawn through the point $(4,\sqrt{3})$ to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1.$ The points at which these tangents touch the ellipse are

1. 
2. 
3. 
4. 

Q. An ellipse whose major axis is parallel to $x-$axis is such that the segments of a focal chord are of $1$ and $3$ units. The lines $ax+by+c=0$ are the chords of the ellipse such that $a,b,c$ are in $A.P.$ and bisected by the point at which they are concurrent. The equation of auxiliary circle is
$x^2+y^2+2\alpha x+2\beta y-2\alpha -1=0.$ Then

1. The locus of perpendicular tangents to the ellipse is $(x-1{)}^2+(y+2{)}^2=7$
2. The locus of perpendicular tangents to the ellipse is $(x-1{)}^2+(y+2{)}^2=19$
3. Area of auxiliary circle is $2\pi$ sq. unit
4. Eccentricity of the ellipse is $\frac{1}{2}$

Q. A common tangent of the two circles is

1. $x+\sqrt{3}y=4$
2. $x=4$
3. $x+2\sqrt{2}y=6$
4. $y=2$

Q. Let the equation $x^2-y^2-2x+2y=0$ represent the pair of tangents to the circle $2x^2-4x+2y^2+1=0$. If $a,L$ represent the lengths of tangents and equation of common chord respectively, then

1. $L:2y=1$
2. $a=1$ units
3. $a=\frac{1}{\sqrt{2}}$ units
4. $L:y=1$

Q.

A set of parallel chords of the parabola $y^2=4ax$ have their mid points on

1. any straight line through the vertex

2. any straight line through the focus

3. a straight line parallel to the axis

4. another parabola

Q. A tangent is drawn to the ellipse $x^2+27y^2=27$ at a point $P\left(\theta \right)$ where $\theta \in (0,\frac{\pi }{2}).$ Then the minimum sum of the intercepts made by the tangent at $P$ on the co-ordinate axes is equal to

Q. The points on the curve $y^2=4a(x+a\sin \frac{x}{a})$ at which the tangent is parallel to x axis lie on -

1. a straight line
2. a parabola
3. an ellipse
4. a circle

Q.

Find the equation of pair of tangents to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ from (5, 4).

1. 5x + 4y + xy = 61

2. 4x + 5y - xy = 21

3. 4x + 5y - xy = 20

4. 4x + 5y + xy = 40

Q. If the tangent at $(1,1)$ on $y^2=x(2-x^2)$ meets the curve again at $P$, then $P$ is

1. $(4,4)$
2. $(-1,2)$
3. $Noneofthese$
4. $(9/4,3/8)$

Q. Find the locus of the middle points of the chords of the circle $x^2+y^2=a^2$ which subtend a right angle at the centre.

Q.

Find the equation of pair of tangents to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ from (5, 4).

1. 4x + 5y - xy = 20

2. 4x + 5y - xy = 21

3. 4x + 5y + xy = 40

4. 5x + 4y + xy = 61

Q. For some parameter R > 0, the curves $y_1=x^2$ and $y_2=Rlnx$ touch each other and have no other point of intersection. Let A be the area bounded between the two curves and y-axis. Then

1. $R=\sqrt{e}$
2. $A=-\frac{8}{3}e\sqrt{e}$
3. $R=2e$
4. $A=\frac{\sqrt{e}}{6}(2e-3)+1$

Q. Locus of the intersection of the tangents at the ends of the normal chords of the parabola $y^2=4ax$ is

1. $(2a+x)y^2+4a^3=0$
2. $(x+2a)y^2+4a^2=0$
3. $(y+2a)x^2+4a^3=0$
4. None of these

Q. Let the pair of tangents from $P(3,4)$ touch the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ at $A(\alpha ,0)$ and $B(\beta ,\gamma )$. If $G$ is the centroid of $△PAB$ and the area of $△GAB$ is equal to $k$ sq. units, then the value of $5k$ is

Q. The locus of point of intersection of pair of tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ if the sum of ordinates of their point of contact is half the length of minor axis, is

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{2x}{a}=0$
2. $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{2y}{b}=0$
3. $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{2x}{a}=0$
4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{2y}{b}=0$

Q. Tangents are drawn through the point $(4,\sqrt{3})$ to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1.$ The points at which these tangents touch the ellipse are

1. $(2,\frac{3\sqrt{3}}{2}),(4,0)$
2. $(2,\frac{\sqrt{3}}{\sqrt{2}}),(4,\frac{\sqrt{3}}{2})$
3. $(4,\frac{3\sqrt{3}}{\sqrt{3}}),(2,0)$
4. $(2,0),(4,0)$

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.

Find the equation of pair of tangents to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ from (5, 4)

1. $4x+5y-xy=20$

2. $4x+5y-xy=21$

3. $4x+5y+xy=40$

4. $5x+4y+xy=2$

Q. An ellipse whose major axis is parallel to $x-$axis is such that the segments of a focal chord are of $1$ and $3$ units. The lines $ax+by+c=0$ are the chords of the ellipse such that $a,b,c$ are in A.P. and bisected by the point at which they are concurrent. The equation of auxiliary circle is $x^2+y^2+2\alpha x+2\beta y-2\alpha -1=0.$ Then

1. The locus of perpendicular tangents to the ellipse is $(x-1{)}^2+(y+2{)}^2=7$
2. The locus of perpendicular tangents to the ellipse is $(x-1{)}^2+(y+2{)}^2=19$
3. Area of auxiliary circle is $2\pi$ sq. unit
4. Eccentricity of the ellipse is $\frac{1}{2}$

Q.

Find the equation of pair of tangents to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ from (5, 4)

1. 4x + 5y - xy = 20

2. 4x + 5y + xy = 40

3. 5x + 4y + xy = 61

4. 4x + 5y - xy = 21