Point Form of Tangent: Ellipse

Q. The equation(s) of the standard ellipse passing through the point (4, -1) and touching the line x + 4y - 10 = 0 is/are:

1. $\frac{x^2}{20}+\frac{y^2}{5}=1$
2. $\frac{x^2}{5}+\frac{y^2}{20}=1$
3. $\frac{x^2}{80}+\frac{4y^2}{5}=1$
4. $\frac{x^2}{5}+\frac{y^2}{80}=1$

Q. The equation of the tangents to the ellipse $3x^2+4y^2=12$ which are parallel to the line $2x-y+5=0$ are

1. $6x-2y\pm \sqrt{\frac{155}{3}}=0$
2. $2x-y\pm \sqrt{19}=0$
3. $16x+22y\pm \sqrt{\frac{155}{3}}=0$
4. $2x+2y\pm \sqrt{39}=0$

Q. The equations of the tangents to the ellipse $x^2+16y^2=16,$ each one of which makes an angle of ${60}^{\circ }$ with the $x-$axis, is

1. $y=\sqrt{3}x\pm 1$
2. $y=\sqrt{3}x\pm 3$
3. $y=\sqrt{3}x\pm 5$
4. $y=\sqrt{3}x\pm 7$

Q. The tangents at the extremities of a latus rectum of an ellipse will intersect at the corresponding directrix.

1. False
2. True

Q.

Let $E_1$ and $E_2$ be two ellipse whose centres are at the origin. The major axes of $E_1$ and $E_2$ lie along x-axis and y-axis respectively. Let S be the circle $x^2+(y-1{)}^2=2$ the straight line $x+y=3$ touches the curves S, $E_1$ and $E_2$ at P, q and R, respectively.
Suppose that $PQ=PR=\frac{2\sqrt{2}}{3}.$ If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$ respectively, then the correct expression(s) is/are

1. ${e_1}^2+{e_2}^2=\frac{43}{40}$

2. $e_1{e}_2=\frac{\sqrt{7}}{2\sqrt{10}}$

3. $|{e_1}^2-{e_2}^2|=\frac{5}{8}$

4. $e_1{e}_2=\frac{\sqrt{3}}{4}$

Q.

If the line $3x+4y=12$ is a tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=2$ then find the point of contact.

1. (3, 4)

2. (6, 8)

3. (8, 6)

4. (4, 8)

Q. A tangent to $3x^2+4y^2=12$ is equally inclined with the coordinate axes. If $d$ is the perpendicular distance from the centre of the ellipse to this tangent, then $2d^2=$ units

Q. If $l_1$ is the tangent to $2x^2+3y^2=35$ at $(4,-1)$ and $l_2$ is the tangent to $4x^2+y^2=25$ at $(2,-3),$ then distance between $l_1$ and $l_2$ is :

1. $\frac{10}{\sqrt{73}}$ unit
2. $\frac{60}{\sqrt{73}}$ unit
3. $\frac{25}{3\sqrt{2}}$ unit
4. $\frac{5}{3\sqrt{2}}$ unit

Q. If the line $x-2y=12$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $(3,-\frac{9}{2})$, then the length of the latus rectum of ellipse is :

1. $9$
2. $8\sqrt{3}$
3. $5$
4. $12\sqrt{2}$

Q. The point of intersection of the tangents at the point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ and its corresponding point $Q$ on the auxiliary circle, meet on the line

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

Q. The equation of tangents to the ellipse $9x^2+16y^2=144$ at the ends of the latus rectum are

1. $\sqrt{7}x+4y=16$
2. $\sqrt{7}x-4y=16$
3. $\sqrt{7}x-4y+16=0$
4. $\sqrt{7}x+4y+16=0$

Q. The equation of the tangents to the ellipse $3x^2+4y^2=12$ which are parallel to the line $2x-y+5=0$ are

1. $6x-2y\pm \sqrt{\frac{155}{3}}=0$
2. $2x-y\pm \sqrt{19}=0$
3. $16x+22y\pm \sqrt{\frac{155}{3}}=0$
4. $2x+2y\pm \sqrt{39}=0$

Q.

The area (in sq units) of the quadrilateral formed by the tangents at the endpoints of the latus recta to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ is___ .

1. $\frac{27}{4}$

2. 18

3. $\frac{27}{2}$

4. 27

Q. The equations of the tangnets to the ellipse $4x^2+3y^2=5$ which are perpendicular to the line $3x-y+7=0$ are :

1. $2x-2y\pm \sqrt{55}=0$
2. $2x+2y\pm \sqrt{55}=0$
3. $2x+6y\pm \sqrt{65}=0$
4. $2x+2y\pm \sqrt{15}=0$

Q. The equation of tangents to the ellipse $9x^2+16y^2=144$ at the ends of the latus rectum are

1. $\sqrt{7}x+4y=16$
2. $\sqrt{7}x-4y=16$
3. $\sqrt{7}x-4y+16=0$
4. $\sqrt{7}x+4y+16=0$

Q. The equations of the tangents to the ellipse $x^2+16y^2=16,$ each one of which makes an angle of ${60}^{\circ }$ with the $x-$axis, is

1. $y=\sqrt{3}x\pm 1$
2. $y=\sqrt{3}x\pm 3$
3. $y=\sqrt{3}x\pm 5$
4. $y=\sqrt{3}x\pm 7$

Q. If the line $\frac{x}{a}+\frac{y}{b}=\sqrt{2}$ touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then the eccentric angle of point of contact is

1. ${45}^{\circ }$
2. ${90}^{\circ }$
3. ${30}^{\circ }$
4. $0^{\circ }$

Q. If $x+y\sqrt{2}=2\sqrt{2}$ is a tangent to the ellipse $x^2+2y^2=4,$ then the eccentric angle of the point of contact is

1. $\frac{\pi }{6}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q. If $l_1$ is the tangent to $2x^2+3y^2=35$ at $(4,-1)$ and $l_2$ is the tangent to $4x^2+y^2=25$ at $(2,-3),$ then distance between $l_1$ and $l_2$ is :

1. $\frac{10}{\sqrt{73}}$ unit
2. $\frac{60}{\sqrt{73}}$ unit
3. $\frac{25}{3\sqrt{2}}$ unit
4. $\frac{5}{3\sqrt{2}}$ unit

Q. The equations of the tangnets to the ellipse $4x^2+3y^2=5$ which are perpendicular to the line $3x-y+7=0$ are :

1. $2x-2y\pm \sqrt{55}=0$
2. $2x+2y\pm \sqrt{55}=0$
3. $2x+6y\pm \sqrt{65}=0$
4. $2x+2y\pm \sqrt{15}=0$

Q. A tangent is drawn to the ellipse $E_1:9x^2+y^2=36$ to cut the ellipse $E_2:3x^2+y^2=48$ at the points $A$ and $B$. If the tangents at $A$ and $B$ to the ellipse $E_2$ intersect at $C(p,3),p>0,$ then the value of $\left[p\right]$ is ($[.]$ represents the greatest integer function)