Parametric Form of Tangent: Ellipse

Q. Angle subtended by common tangents intercepted between two ellipses $4(x-4{)}^2+25y^2=100$ and $4(x+1{)}^2+y^2=4$ at origin is

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

Q. Common tangents are drawn to the parabola $y^2=4x$ and the ellipse $3x^2+8y^2=48$ touching the parabola $A$ and $B$ and the ellipse at $C$ and $D$. Area of quadrilateral $ABCD$ (in sq.units) is:

1. $55\sqrt{2}$
2. $50$
3. $50\sqrt{2}$
4. $100\sqrt{2}$

Q.

Tangent is drawn to ellipse $\frac{x^2}{27}+y^2=1at$
$(3\sqrt{3}cos\theta ,sin\theta )$ $(where,\theta \in (0,\frac{\pi }{2}))$.
Then, the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum, is

1. $\frac{\pi }{3}$

2. $\frac{\pi }{8}$

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

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

Q. If normal at $P(2,\frac{3\sqrt{3}}{2})$ meets the major axis of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ at $Q$ and $S,S^{\prime }$ are foci of given ellipse along positive and negative directions of axes, then the ratio $SQ:S^{\prime }Q$ is

1. $\frac{7-\sqrt{8}}{7+\sqrt{8}}$
2. $\frac{8-\sqrt{7}}{8+\sqrt{7}}$
3. $\frac{7+\sqrt{8}}{7-\sqrt{8}}$
4. $\frac{8+\sqrt{7}}{8-\sqrt{7}}$

Q. A vertical line passing through the point $(h,0)$ intersects the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta \left(h\right)=\text{area of the triangle}PQR$, ${\Delta }_1=\underset{1/2\le h\le 1}{max}\Delta \left(h\right)$ and ${\Delta }_2=\underset{1/2\le h\le 1}{min}\Delta \left(h\right)$, then $\frac{8}{\sqrt{5}}{\Delta }_1-8{\Delta }_2=$

Q.

If angle $\theta$ is divided into two parts such that the tangents of one part is $\lambda$ times the tangent of other, and $\varphi$ is their difference, then show that sin $\theta =\frac{\lambda +1}{\lambda -1}sin\varphi$.

Q.

If $a>2b>0$ then the positive value of m for which $y=mx-b\sqrt{1+m^2}$ is a common tangent to $x^2+y^2=b^2$and${(x-a)}^2+y^2=b^2$ is

1. $\frac{2b}{\sqrt{a^2-4b^2}}$

2. $\frac{\sqrt{a^2-4b^2}}{2b}$

3. $\frac{2b}{a-2b}$

4. $\frac{b}{a-2b}$

Q. The tangent at any point on the curve $x=a{\cos }^3\theta ,y=a{\sin }^3\theta$ meets the coordinate axes at $P$ and $Q$.
The locus of the mid point of $PQ$ is

1. $x^2+y^2=a^2$
2. $2(x^2+y^2)=a^2$
3. $4(x^2+y^2)=a^2$
4. $(x^2+y^2)=4a^2$

Q. If $p,p^{\prime }$ denote the lengths of the perpendiculars from the focus and the centre of an ellipse whose semi major axis is of length $a$ units on a tangent at a point on the ellipse and $r$ denotes the focal distance of the point, then

1. $rp=a{p}^{\prime }$
2. $rp+1=a{p}^{\prime }$
3. $ap=r{p}^{\prime }-1$
4. $ap=r{p}^{\prime }$

Q. If $p$ is the length of the perpendicular from focus upon the tangent at any point $P$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $r$ is the distance of $P$ from the focus, then $(\frac{2a}{r}-\frac{b^2}{p^2})$ is equal to

Q. Let $P_i$ and $P_i^{\prime }$ be the feet of perpendiculars drawn from foci $S,S^{\prime }$ on a tangent $T_i$ to an ellipse whose length of semi major axis is $20,$ if $\sum _{i=1}^{10}\left(S{P}_i\right)\left(S{P}_i^{\prime }\right)=2560,$ then the value of eccentricity is

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

Q. The tangent at $\alpha$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the auxiliary circle at two points which subtend a right angle at the centre. Then e =

1. 
2. 
3. 
4. 

Q. The slopes of the common tangents of the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ and circle $x^2+y^2=3$ are

1. $\pm 1$
2. $\pm \sqrt{2}$
3. $\pm \sqrt{3}$
4. None of these

Q. If the tangent at point P on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major axis at T, C is the centre and PN is the perpendicular on major axis, then the value of CN.CT is

1. 
2. 
3. 
4. 

Q.

If $CF$ is perpendicular from the centre $C$ of the ellipse $\frac{x^2}{49}+\frac{y^2}{25}=1$ on the tangent at any point $P$, and $G$ is the point where th e normal at $P$ meets the minor axis, then $(CF\cdot PG{)}^2$ is equal to___

Q. The value(s) of $\left[c\right]$ for which the line $y=4x+c$ touches the curve $x^2+16y^2=16$ is/are (where $[.]$ represents greatest integer function)

1. $-16$
2. $-17$
3. $16$
4. $17$

Q. Consider a curve $a{x}^2+2hxy+b{y}^2=1$ and a point P not on the curve. A line is drawn from the point P and intersects the curve at points Q and R. If the product $PQ\times QR$ is independent of the slope of the line then

1. equation of the curve is a circle with radius $\frac{1}{\sqrt{a}}$
2. locus of $P$ is a straight line with $x$-intercept $a$
3. equation of the curve is a circle with radius $\frac{1}{\sqrt{b}}$
4. locus of $P$ is a straight line with $x$-intercept $b$

Q. An ellipse is rotated through a right angle in its own plane about its centre, which is fixed. If at a point tangents are drawn before and after rotation then locus of point of intersection of such tangents is

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

Q. If the tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ at the point $(a\cos \theta ,b\sin \theta )$ meets the auxiliary circle in two points $A,B$ such that the chord $AB$ subtends a right angle at the centre, then the eccentricity of the ellipse is given by $\frac{1}{\sqrt{\alpha +\beta {\sin }^2\theta }},$ then the value of $(\alpha +\beta {)}^2$ is equal to

Q.

The line drawn from (4, -1, 2) to the point (-3, 2, 3) meets a plane at right angles at the point (-10, 5, 4), then the equation of plane is
[DSSE 1985]

1. 
   

2. 
   

3. 
   

4. None of these

Q. If the tangent at $\theta$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the auxiliary circle at two points which subtend a right angle at the centre, then $e^2(2-{\cos }^2\theta )=$

1. 
2. 
3. 
4. 

Q. The value(s) of $\left[c\right]$ for which the line $y=4x+c$ touches the curve $x^2+16y^2=16$ is/are (where $[.]$ represents greatest integer function)

1. $-16$
2. $-17$
3. $16$
4. $17$

Q. An ellipse is rotated through a right angle in its own plane about its centre, which is fixed. If at a point tangents are drawn before and after rotation then locus of point of intersection of such tangents is

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

Q. If the focus of a parabola is (-2, 1) and the directrix has the equation x+y=3, then the vertex is:

1. 
2. 
3. 
4. 

Q. Let image of a variable point $(–4\cos \theta ,–3\sin \theta )$ about the line $x+y=0$ lie on curve $C$ and pair of straight lines represented by the equation $m(x^2–y^2)+(m^2–1)xy=0$ denotes two chords of the curve $C$ for $m\in R-\left\{0\right\}.$ Then choose the correct option(s).

1. If perpendiculars are drawn on the lines joining end points of the chords from the origin, then locus of feet of these perpendiculars is $x^2+y^2=\frac{144}{25}.$
2. Minimum length of the segment joining end points of the given pair of chords is $\frac{24}{5}.$
3. Product of the length of perpendiculars drawn from the point $(0,\sqrt{7})$ and $(0,-\sqrt{7})$ upon a variable tangent to curve $C$ is $9.$
4. Product of the length of perpendiculars drawn from the point $(0,\sqrt{7})$ and $(0,-\sqrt{7})$ upon a variable tangent to curve $C$ is $25.$

Q. Tangents are drawn from any point on the circle $x^2+y^2+2gx+2fy+c=0$ to the circle $x^2+y^2+2gx+2fy+csi{n}^2\alpha +(g^2+f^2)co{s}^2\alpha =0$, then the angle between the tangents

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

Q. Let $L$ be a common tangent line to the curves $4x^2+9y^2=36$ and $(2x{)}^2+(2y{)}^2=31.$ Then the square of the slope of the line $L$ is

Q. Show that the lines $\frac{x+4}{3}=\frac{y+6}{5}=\frac{z-1}{-2}$ and 3x − 2y + z + 5 = 0 = 2x + 3y + 4z − 4 intersect. Find the equation of the plane in which they lie and also their point of intersection.

Q. Circles $C_1$ and $C_2$ , of radii $r$ and $R$ respectively, touch each other as shown in the figure. The line $A$, which is parallel to the line joining the centres of $C_1$ and $C_2$ , is tangent to $C_1$ at $P$ and intersects $C_2$ at $A,B$. If $R^2=2r^2$, then $\angle AOB$ equals

(correct answer + 2, wrong answer - 0.50)

1. $22{\frac{1}{2}}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. $67{\frac{1}{2}}^{\circ }$

Q. Slope of the common tangent to the ellipses $\frac{x^2}{a^2+b^2}+\frac{y^2}{b^2}=1,\frac{x^2}{a^2}+\frac{y^2}{a^2+b^2}=1$ is:

1. $\frac{b}{a}$
2. $\frac{a}{b}$
3. $\frac{2a}{b}$
4. $\frac{2b}{a}$