Slope Form of Tangent: Ellipse

Q. Tangents are drawn to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ at the ends of both latus rectum. The area of the quadrilateral so formed is

1. $27$ sq. units
2. $\frac{13}{2}$ sq. units
3. $\frac{15}{4}$ sq. units
4. $45$ sq. units

Q.

Find the condition for the line $x\cos \alpha +y\sin \alpha =p$ to be a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Q. A ray of light through $(2,1)$ is reflected at a point $P$ on the $y-$axis and then passes through the point $(5,3)$. If this reflected ray is the directrix of an ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{\sqrt{53}},$ then the equation of the other directrix can be

1. $2x-7y+29=0$ or $2x-7y-7=0$
2. $11x+7y+8=0$ or $11x+7y-15=0$
3. $2x-7y-39=0$ or $2x-7y-7=0$
4. $11x-7y-8=0$ or $11x+7y+15=0$

Q. The point $P(-2\sqrt{6},\sqrt{3})$ lies on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ having eccentricity $\frac{\sqrt{5}}{2}.$ If the tangent and normal at $P$ to the hyperbola intersect its conjugate axis at the points $Q$ and $R$ respectively, then $QR$ is equal to

1. $3\sqrt{6}$
2. $6$
3. $6\sqrt{3}$
4. $4\sqrt{3}$

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. The equations of the tangents to the ellipse $9x^2+16y^2=144$ from the point (2, 3) are

1. x=2, y=3
   
2. y=3, x=5
   
3. x=3, y=2
   
4. x+y=5, 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. 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. If the complex number $z_1,z_2$ the origin form an equilateral triangle then $z_1^2+z_2^2$ =

1. 
2. 
3. 
4. 

Q. Tangents are drawn from the point $(4,2)$ to the curve $x^2+9y^2=9,$ then the tangent of acute angle between the tangents is

1. $\frac{\sqrt{43}}{5}$
2. $\frac{3\sqrt{3}}{5\sqrt{17}}$
3. $\sqrt{\frac{3}{17}}$
4. $\frac{\sqrt{43}}{10}$

Q. If two concentric ellipses be such that the foci of one be on the other and if $\frac{\sqrt{3}}{2}$ and $\frac{1}{\sqrt{2}}$ be their eccentricities. Then angle between their axes is

1. ${\cos }^{-1}\frac{1}{\sqrt{6}}$
2. ${\cos }^{-1}\frac{2}{3\sqrt{3}}$
3. ${\cos }^{-1}\frac{\sqrt{2}}{3}$
4. ${\cos }^{-1}\sqrt{\frac{2}{3}}$

Q. The equations of tangents drawn from the point (2, 3) to the ellipse $9x^2+16y^2=144$ are:

1. $y=3$
2. $y=5$
3. $y=-x+3$
4. $y=-x+5$

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

Q. $B$ is an extremity of the minor axis of the ellipse whose foci are $S$ and $S^{\prime }.$ If $\angle SB{S}^{\prime }$ is a right angle, then eccentricity of the ellipse is

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

Q. If two tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ make angles $\alpha$ and $\beta$ with the major axis such that $\tan \alpha +\tan \beta =\lambda ,$ then the locus of their point of intersection is

1. $x^2+y^2=a^2$
2. $x^2+y^2=b^2$
3. $x^2-a^2=2\lambda xy$
4. $\lambda (x^2-a^2)=2xy$

Q. If line $x+y=3$ is a tangent to the ellipse with foci at $(4,3)$ and $(6,k)$ at point $(1,2),$ then the value of $k$ is

1. $27$
2. $17$
3. $20$
4. $15$

Q. If tangents $PQ$ and $PR$ are drawn from a point on the circle $x^2+y^2=16$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{12}=1(a<3)$, so that the fourth vertex $S$ of the parallelogram $PQSR$ lies on the circumcircle of the triangle $PQR$, then the eccentricity of the ellipse is

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

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

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.

The angle between tangents to the curves $y=x^2$ and $x=y^2$ at $\left(1,1\right)$ is :

1. $0$

2. ${\tan }^{-1}1$

3. ${\tan }^{-1}\left(\frac{3}{4}\right)$

4. ${\tan }^{-1}\left(\frac{1}{3}\right)$

Q.

Focus of hyperbola is $(\pm 3,0)$ and equation of tangent is $2x+y-4=0$, find the equation of hyperbola is

1. $4x^2–5y^2=20$

2. $5x^2–4y^2=20$

3. $4x^2–5y^2=1$

4. $5x^2–4y^2=1$

Q. The equations of the tangents to the hyperbola $3x^2-4y^2=12$ which are parallel to the line 2x + y + 7 = 0 are

1. 
2. 
3. 
4. 

Q. Distance between the points on the curve $4x^2+9y^2=1,$ where tangent is parallel to the line $8x=9y,$ is $\frac{2}{\sqrt{k}},$ unit, then $k=$

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

Q. Consider the parabola whose focus is at $(0,0)$ and tangent at vertex is $x-y+1=0.$ Then

1. the length of latus rectum is $2\sqrt{2}$
2. the length of the chord of parabola on the $x$-axis is $4\sqrt{2}$
3. equation of directrix is $x-y-2=0$
4. tangents drawn to the parabola at the extremities of the chord $3x+2y=0$ intersect at an angle of $\frac{\pi }{3}$

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

Q. If line $x+y=3$ is a tangent to the ellipse with foci at $(4,3)$ and $(6,k)$ at point $(1,2),$ then the value of $k$ is

1. $27$
2. $17$
3. $20$
4. $15$