Position of a Line W.R.T Ellipse

Q. Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l² + m² − n² = 0
(ii) 2l − m + 2n = 0 and mn + nl + lm = 0
(iii) l + 2m + 3n = 0 and 3lm − 4ln + mn = 0
(iv) 2l + 2m − n = 0, mn + ln + lm = 0

Q. The value(s) of $\lambda$ for which the line $y=x+\lambda$ touches the ellipse
$9x^2+16y^2=144$ is/are:

1. $5$
2. $-5$
3. $10$
4. $-10$

Q. The asymptote of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ forms a triangle with any tangent to the hyperbola. The area of such triangle formed is $a^2\tan \lambda$ , where $\lambda \in (0,\frac{\pi }{2})$ then its eccentricity is

1. $\sec \lambda$
2. $\text{cosec}\lambda$
3. ${\sec }^2\lambda$
4. ${\text{cosec}}^2\lambda$

Q. The minimum area of a triangle formed by the tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and coordinate axes is

1. $\frac{(a+b{)}^2}{2}$ sq. units
2. $ab$ sq. units
3. $\frac{a^2+b^2}{2}$ sq. units
4. $\frac{a^2+ab+b^2}{2}$ sq. units

Q. Let $T$ be the tangent to the ellipse $E:x^2+4y^2=5$ at the point $P(1,1).$ If the area of the region bounded by the tangent $T$, ellipse $E$, lines $x=1$ and $x=\sqrt{5}$ is $\alpha \sqrt{5}+\beta +\gamma {\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)$, then $|\alpha +\beta +\gamma |$ is equal to

Q. A coplanar beam of light emerging from a point source have the equation $\lambda x–y+2(1+\lambda )=0,\forall \lambda \in R$; the rays of the beam strike an elliptical surface and get reflected inside the ellipse. The reflected rays form another convergent beam having the equation $\mu x-y+2(1-\mu )=0,\forall \mu \in R$. Further it is found that the foot of the perpendicular from the point $(2,2)$ upon any tangent to the ellipse lies on the circle $x^2+y^2–4y–5=0$.

1. The eccentricity of the ellipse is equal to $\frac{2}{3}$.
2. The eccentricity of the ellipse is equal to $\frac{1}{3}$
3. The area of the largest triangle that an incident ray and corresponding reflected ray can enclose with major axis of the ellipse is equal to $2\sqrt{5}$.
4. The area of the largest triangle that an incident ray and corresponding reflected ray can enclose with major axis of the ellipse is equal to $4\sqrt{5}$

Q. If the tangent to the ellipse $x^2+4y^2=16$ at the point $P\left(\varphi \right)$ is normal to the circle $x^2+y^2-8x-4y=0,$ then possible values(s) of $\varphi$ is/are

1. $\frac{\pi }{4}$
2. $0$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q. Number of normals that can be drawn from the point (0, 0) to $3x^2+2y^2=30$ are

1. 2
2. 4
3. 1
4. 3

Q. $\mathrm{The}\mathrm{straight}\mathrm{line}\mathrm{lx}+\mathrm{my}+n=0\mathrm{touches}\mathrm{the}\mathrm{ellipse}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\mathrm{if}$

1. $a^2{l}^2+b^2{m}^2=n^2$
2. $a^2{l}^2+b^2{m}^2=n$
3. $\mathrm{al}+\mathrm{bm}=n$
4. $a^2{l}^2+b^2{m}^2=0$

Q. $ABCD$ is a rhombus. The slope of $AC$ is $1$ and is among the family of lines $(x+2y-5)+\lambda (3x+y-5)=0,$ where $\lambda \in R.$. One of the vertex of the rhombus is $(-2,3)$. If the area of rhombus is $10\sqrt{2}$ sq. units, then which of the following is/are correct?

1. Length of smaller diagonal is $4$
2. Length of larger diagonal is $4\sqrt{2}$
3. One of the vertex is $(2,-1)$
4. Perimeter of rhombus is $2\sqrt{57}$

Q. Let $F_1$ and $F_2$ be the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{17}=1$ and $M=|P_i{F}_1–P_i{F}_2|,i=1,2,3,4$ where $P_1,P_2,P_3,P_4$ are four points on the curve $4x^2–4xy+y^2–81=0$ such that either $M$ is greatest or least. If $S$ is set of distances between any $2$ different points of $P_i,$ then $S_{max}+S_{min}=$

Q. For the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ and circle $(x-1{)}^2+(y+2{)}^2=3,$ the centre of the circle lies

1. outside the ellipse
2. on the ellipse
3. none of these
4. inside the ellipse

Q. Area (in sq. units) of the region outside $\frac{|x|}{2}+\frac{|y|}{3}=1$ and inside the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is:

1. $3(\pi -2)$
2. $6(\pi -2)$
3. $6(4-\pi )$
4. $3(4-\pi )$

Q. The angle between the straight lines
$\frac{x+1}{2}=\frac{y-2}{5}=\frac{z+3}{4}\mathrm{and}\frac{x-1}{1}=\frac{y+2}{2}=\frac{z-3}{-3}$ is
(a) 45°
(b) 30°
(c) 60°
(d) 90°

Q. Show that the three lines with direction cosines $\frac{12}{13},\frac{-3}{13},\frac{-4}{13};\frac{4}{13},\frac{12}{13},\frac{3}{13};\frac{3}{13},\frac{-4}{13},\frac{12}{13}$ are mutually perpendicular.

Q. A ray of light is sent along the line $x-2y+5=0$; upon reaching the line $3x-2y+7=0,$ the ray is reflected from it. Find the equation of the line containing the reflected ray.

Q. The minimum area of a triangle formed by the tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and coordinate axes is

1. $ab$ sq. units
2. $\frac{a^2+b^2}{2}$ sq. units
3. $\frac{(a+b{)}^2}{2}$ sq. units
4. $\frac{a^2+ab+b^2}{2}$ sq. units

Q. Write the angle between the lines 2x = 3y = −z and 6x = −y = −4z.

Q. Find the angle between the following pairs of lines:
(i) $\frac{x+4}{3}=\frac{y-1}{5}=\frac{z+3}{4}\mathrm{and}\frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2}$

(ii) $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{-3}\mathrm{and}\frac{x+3}{-1}=\frac{y-5}{8}=\frac{z-1}{4}$

(iii) $\frac{5-x}{-2}=\frac{y+3}{1}=\frac{1-z}{3}\mathrm{and}\frac{x}{3}=\frac{1-y}{-2}=\frac{z+5}{-1}$

(iv) $\frac{x-2}{3}=\frac{y+3}{-2},z=5\mathrm{and}\frac{x+1}{1}=\frac{2y-3}{3}=\frac{z-5}{2}$

(v) $\frac{x-5}{1}=\frac{2y+6}{-2}=\frac{z-3}{1}\mathrm{and}\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-6}{5}$

(vi) $\frac{-x+2}{-2}=\frac{y-1}{7}=\frac{z+3}{-3}\mathrm{and}\frac{x+2}{-1}=\frac{2y-8}{4}=\frac{z-5}{4}$

Q. Let $f\left(x\right)=\{\begin{array}{cc}|x-1|+a,& x\le 1\\ 2x+3,& x>1\end{array}$ If $f\left(x\right)$ has local minimum at $x=1$ and $a\ge 5$ then $a$ is equal to

Q. Let $S$ be a curved mirror passing through $(3,4)$ having the property that all the light emerging from origin(focus) , after getting reflected from the mirror becomes parallel to $x-$axis. The angle between the tangents drawn from the point $(-2,6)$ is ${90}^{\circ }$. If the circle $(x-4{)}^2+y^2=r^2$ internally touches the curve $S,$ then the value of $r^2$ is

Q. The minimum area of triangle formed by the tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and coordinate axes is:

1. $ab$ sq. units
2. $\frac{a^2+b^2}{2}$ sq. units
3. $\frac{a^2+ab+b^2}{3}$sq. units
4. $\frac{(a+b{)}^2}{2}$ sq. units

Q. Let $F_1$ and $F_2$ be the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{17}=1$ and $M=|P_i{F}_1–P_i{F}_2|,i=1,2,3,4$ where $P_1,P_2,P_3,P_4$ are four points on the curve $4x^2–4xy+y^2–81=0$ such that either $M$ is greatest or least. If $S$ is set of distances between any $2$ different points of $P_i,$ then $S_{max}+S_{min}=$

Q. If from any point on the circle $x^2+y^2+2gx+2fy+c=0$ tangents are drawn to the circle $x^2+y^2+2gx+2fy+c{\sin }^2\alpha +(g^2+f^2){\cos }^2\alpha =0$, then the angle between the tangents is

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

Q. Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l² + m² − n² = 0
(ii) 2l − m + 2n = 0 and mn + nl + lm = 0
(iii) l + 2m + 3n = 0 and 3lm − 4ln + mn = 0

Q. Name the type of the quadrilateral formed by joining the points $A(-1,-2),B(1,0),C(-1,2)$ and $D(-3,0)$ on a graph paper. Justify your answer.

Q.

Find the set of values of $\lambda$ for which the
line $3x-4y+\lambda =0$ intersects the ellipse
$\frac{x^2}{16}+\frac{y^2}{a}=1$ at 2 distinct point.

1. $(-12\sqrt{2},12\sqrt{2})$

2. $(-12,12)$

3. $(12,12\sqrt{2})$

4. $(-12,12\sqrt{2})$

Q. The line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the curve $xy=c^2,z=0$ if c is equal to

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

Q. The value of m for which the area of the triangle included between the axes and any tangent to the curve $x^{m}y=b^m$ is constant, is

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

Q. Why was the poet in a dilemma?