Point Form of Tangent: Ellipse

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. 27

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

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

4. 18

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

Q.

Let $P(h,k)$ be a point on the curve $y=x^2+7x+2$, nearest to the line, $y=3x-3.$ Then the equation of the normal to the curve at $P$ is

1. $x+3y-62=0$

2. $x-3y-11=0$

3. $x-3y+22=0$

4. $x+3y+26=0$

Q. If tangents are drawn to the ellipse $x^2+2y^2=2$ at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinates axes lie on the curve :

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

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.

If the parabola $y=a{x}^2-6x+b$ passes through $(0,2)$ and has its tangent at $x=\frac{3}{2}$ parallel to the $\mathrm{x}$-axis then

1. $a=b=0$

2. $a=b=1$

3. $a=b=2$

4. $a=b=-1$

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

Q.

The sum of the squares of perpendicular on any tangent of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$ = 1 from two points on the minor axis is

1. 2a²

2. a²

3. b²

4. 2b²

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.

Locus of the feet of the perpendicular drawn from focus of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ upon any tangent is _____

1. x² + y² = 2a²

2. x² + y² = a²

3. x² - y² = a²

4. x² - y² = 2a²

Q. If the tangent at the point $(4\cos a,\frac{16}{\sqrt{11}}\sin a)$ to the ellipse $16x^2+11y^2=256$ is also a tangent to the circle $x^2+y^2-2x=15$, then the value of $a$ is/are

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{2\pi }{3}$
4. $\frac{5\pi }{3}$

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

Q. The equation of the tangents to the ellipse $3x^2+4y^2=12,$ which are perpendicular to the line $y+2x=4,$ are

1. $y=2x\pm 2$
2. $2y=-x\pm 4$
3. $2y=x\pm 4$
4. $y=-2x\pm 2$

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.

The line $2x+\sqrt{6}y=2$ is a tangent to the curve $x^2-2y^2=4$. The point of contact is

1. $\left(4,-\sqrt{6}\right)$

2. $\left(7,-2\sqrt{6}\right)$

3. $\left(2,3\right)$

4. $\left(\sqrt{6},1\right)$

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

Q. For which of the following curves, the line $x+\sqrt{3}y=2\sqrt{3}$ is the tangent at the point $(\frac{3\sqrt{3}}{2},\frac{1}{2})$ ?

1. $x^2+9y^2=9$
2. $2x^2-18y^2=9$
3. $y^2=\frac{1}{6\sqrt{3}}x$
4. $x^2+y^2=7$

Q. The equation of straight lines which are both tangent and normal to the curve $27x^2=4y^3$ are

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

Q.

If the line $y\cos \alpha =x\sin \alpha +a\cos \alpha$ is a tangent to the circle $x^2+y^2=a^2$, then

1. ${\sin }^2\alpha =1$

2. ${\cos }^2\alpha =1$

3. ${\sin }^2\alpha =a^2$

4. ${\cos }^2\alpha =a^2$

Q. An ellipse is sliding along the coordinate axis. If at a particular instant the foci of the ellipse are at $(1,1)$ and $(3,3),$ then the area of the director circle of the ellipse is

1. $2\pi$ sq. units
2. $4\pi$ sq. units
3. $8\pi$ sq. units
4. $6\pi$ sq. units

Q. Let d and d’ be the perpendicular distances from the foci of an ellipse to the tangent at P on the ellipse, whose foci are S and S’. Then S’P:SP =

Q.

If an equation of a tangent to the curve, $y=\cos (x+y),-1\le x\le 1+\pi$, is $x+2y=k$ then $k$ is equal to :

1. $1$

2. $2$

3. $\frac{\mathrm{\pi }}{2}$

4. $\frac{\mathrm{\pi }}{4}$

Q. The equation of latus rectum of a parabola is x + y = 8 and the equation of the tangent at the vertex is x+y = 12, then length of the latus rectum is

1. 
   
2. 
   
3. 
4. 8

Q. Through the point P(1, 2, 2, ) a plane is drawn at right angles to OP, O being the origin, to meet the axes in A, B, C. If the area of triangle ABC is $\frac{402+\lambda }{8}$ then $\lambda$ is___

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

Q. The equation of the tangents to the ellipse $3x^2+4y^2=12,$ which are perpendicular to the line $y+2x=4,$ are

1. $2y=x\pm 4$
2. $y=2x\pm 2$
3. $2y=-x\pm 4$
4. $y=-2x\pm 2$

Q. Let $E_1$ and $E_2$ be the two ellipses centred at origin. The major axis of $E_1$ and $E_2$ lie along the $x-$ axis and $y-$ axis respectively. Let $S$ be the circle $x^2+(y-1{)}^2=2,$ the straight line $x+y=3$ touches the curve $S,E_1$ and $E_2$ at $P,Q$ and $R$ respectively such that $PQ=PR=\frac{2\sqrt{2}}{3}.$ If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2,$ then which of the following is/are correct

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 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. $2x+2y\pm \sqrt{39}=0$