Point Form of Tangent: Ellipse

Q.

The equation of tangent to the conic $S\equiv a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0at(x_1,y_1)$ is obtained by replacing x with $\frac{x+x_1}{2},ywith\frac{y+y_1}{2},$with $x^2$ with $x{x}_1,y^2$ with $y{y}_1,xy$ with $\frac{x{y}_1+y{x}_1}{2}$ and keeping c as it is in the equation of conic (S).

1. True

2. False

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

1. False
2. True

Q. The sum of the squares of perpendicuars on any tangents of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ from two points on minor axis each one at a distance of $\sqrt{a^2-b^2}$ unit from the centre is

1. $2b^2$
2. $2a^2$
3. $a^2$
4. $b^2$

Q. A tangent having slope of $-\frac{4}{3}$ to the ellipse $\frac{x^2}{18}+\frac{y^2}{32}=1$ intersects the major and minor axes at point $A$ and $B$ respectively. If $C$ is the centre of the ellipse, then the area of the triangle $ABC$ (in sq. unit) is

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. The coordinates $(2,3)$ and $(1,5)$ are the foci of an ellipse which passes through the origin, then the equation of

1. Tangent at origin is $(3\sqrt{2}-5)x+(1-2\sqrt{2})y=0$
2. Tangent at origin is $(3\sqrt{2}+5)x+(1+2\sqrt{2})y=0$
3. Normal at the origin is $(3\sqrt{2}+5)x-(1+2\sqrt{2})y=0$
4. Normal at the origin is $x(3\sqrt{2}-5)-y(1-2\sqrt{2})=0$

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)

Q. Find the equation of tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ at the point whose eccentric angle is $\frac{4\pi }{3}$

1. $4x+5\sqrt{3}y-40=0$
2. None of the above
3. $4x+5\sqrt{3}y+40=0$
4. $8x+5\sqrt{3}y+80=0$

Q.

Let $F_1(x_1,0)$ and $F_2(x_2,0),$ where, $x_1<0$ 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 the parabola at M meets the X-axis at Q then the ratio of area of $\Delta MQR$ to area of the quadrilateral $M{F}_{1}N{F}_2$ is

1. 3:4

2. 4:5

3. 5:8

4. 2:3