Homogenization

Q. Let $PQ$ be a chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which subtends right angle at the centre $(0,0).$ Then its distance from the centre is equal to

1. $\frac{ab}{\sqrt{a^2+b^2}}$
2. $\frac{a}{\sqrt{a^2+b^2}}$
3. $\frac{b}{\sqrt{a^2+b^2}}$
4. $\frac{2ab}{\sqrt{a^2+b^2}}$

Q. If the angle between the two lines represented by $2x^2+5xy+3y^2+6x+7y+4=0$ is ${\tan }^{-1}\left(m\right),$ then the value(s) of $m$ is/are

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

Q. If the equation $6x^2-\alpha xy-3y^2-24x+3y+\beta =0$ represents a pair of straight lines that intersect on the $x-$axis, then the value of $20\alpha -\beta$ is

Q.

The lines represented by the equation $9x^2+24xy+16y^2+21x+28y+6=0$ are

1. Parallel

2. Coincident

3. Perpendicular

4. None of these

Q. If the solution curve of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passes through the points $(2,1)$ and $(k+1,2),k>0$, then

Q. If the angle between the pair of straight lines formed by joining the points of intersection of $x^2+y^2=4$ and $y=3x+c$ to the origin is right angle, then $c^2=$

Q. Points $A$ and $B$ lie on the auxiliary circle of ellipse $\frac{x^4}{4}+y^2=1.$ $P$ and $Q$ are the corresponding points on the ellipse for the points $A$ and $B$ respectively ($O$ is the origin). Which of the following options is/are CORRECT?

1. The maximum value of angle $AOP$ is ${\tan }^{-1}\left(2\sqrt{2}\right)$
2. The maximum value of angle $AOP$ is ${\tan }^{-1}\left(\frac{1}{2\sqrt{2}}\right)$
3. If $OA\perp OB$ and $Q^{\prime }$ is reflection of $Q$ in origin, then the minimum value of angle $PO{Q}^{\prime }$ is ${\tan }^{-1}\left(\frac{4}{3}\right)$
4. If $OA\perp OB$ and $Q^{\prime }$ is reflection of $Q$ in origin, then the minimum value of angle $PO{Q}^{\prime }$ is ${\tan }^{-1}\left(\frac{3}{4}\right)$

Q. Let $(x,y)\in Q_1,2x\le y$ be such that
${\sin }^{-1}\left(ax\right)+{\cos }^{-1}\left(y\right)+{\cos }^{-1}\left(bxy\right)=\frac{\pi }{2}.$
Match the statements in Column I with the locus in Column II.
$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \left(A\right)& \text{If}a=1\text{and}b=0,\text{then}(x,y)& \left(p\right)& \text{lies on part of the circle}{x}^2+y^2=1\\ \left(B\right)& \text{If}a=1\text{and}b=1,\text{then}(x,y)& \left(q\right)& \text{lies on part of the curve represented by}(x^2-1)(y^2-1)=0\\ \left(C\right)& \text{If}a=1\text{and}b=2,\text{then}(x,y)& \left(r\right)& \text{lies on part of the line}y=x\\ \left(D\right)& \text{If}a=2\text{and}b=2,\text{then}(x,y)& \left(s\right)& \text{lies on part of the curve represented by}(4x^2-1)(y^2-1)=0\end{array}$

1. $A\to p,B\to q,C\to p,D\to s$
2. $A\to p,B\to q,C\to p,D\to r$
3. $A\to p,B\to q,C\to r,D\to s$
4. $A\to r,B\to q,C\to p,D\to s$

Q. If the lines represented by $a{x}^2+8xy+3y^2=0$ are mutually perpendicular, then

1. $x-3y=0$ is one of the line representing pair of lines
2. $|a|=3$
3. $y-3x=0$ is one of the line representing pair of lines
4. $|a|=\frac{8}{3}$

Q. If the line $\frac{x}{a}+\frac{y}{b}=1$ intersects the curve
$5x^2+5y^2+5bx+5ay-9ab=0$ at $P$ and $Q$ such that $\angle POQ={90}^{\circ },$ where $O$ is the origin, then the value of $\frac{a}{b}$ is

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

Q. The equation of the lines represented by $4x^2+24xy+11y^2=0$ is/are

1. $2x+y=0$
2. $2x+11y=0$
3. $x-y=0$
4. $y-2x=0$

Q. The angle between the lines joining the origin to the points of intersection of the line $y=3x+2$ with the curve $x^2+2xy+3y^2+4x+8y=11,$ is

1. $\frac{1}{2}{\tan }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$
2. ${\tan }^{-1}\left(\frac{\sqrt{2}}{3}\right)$
3. ${\tan }^{-1}\left(\frac{\sqrt{3}}{2\sqrt{2}}\right)$
4. ${\tan }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$

Q. Find the angle between the two straight lines whose direction cosines $l,m,n$ are given by $2l+2m-n=0$ and $mn+nl+lm=0$

Q. If the angle between the two lines represented by $2x^2+5xy+3y^2+6x+7y+4=0$ is ${\tan }^{-1}\left(m\right),$ then the value(s) of $m$ is/are

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

Q. The asymptotes of the curve $x^2+4xy+3y^2+4x-3y+1=0$ passes through a fixed point $(h,k)$ then $h+k$ is

Q. $t_1$ and $t_2$ are two points on the parabola $y^2=4x$. If the chord joining them is a normal to the parabola at $t_1$ then

1. $t_1+t_2=0$
2. $t_1(t_1+t_2)=1$
3. $t_1(t_1+t_2)+2=0$
4. $t_1{t}_2+1=0$

Q. The foot of the perpendicular drawn from the origin, on the line, $3x+y=\lambda (\lambda \ne 0)$ is $P$. If the line meets x-axis at $A$ and y-axis at $B$, then the ratio $BP:PA$ is

1. $9:1$
2. $1:3$
3. $3:1$
4. $1:9$

Q. There are $3$ coplanar (non-concurrent) lines $A,B$ and $C$, from these lines $10,12,11$ points are chosen respectively, such that no points other than points lying on the same line are collinear, then the total number of triangles that can be formed using these points are

1. ${}^{32}{C}_2+{}^{32}{C}_3-({}^{10}{C}_3+{}^{12}{C}_3+{}^{11}{C}_3)$
2. ${}^{33}{C}_3-{(}^{11}{C}_4+{}^{12}{C}_3)$
3. ${}^{10}{C}_2\times {}^{23}{C}_1{+}^{11}{C}_2\times {}^{22}{C}_1{+}^{12}{C}_2\times {}^{21}{C}_1{+}^{10}{C}_1\times {}^{11}{C}_1\times {}^{12}{C}_1$
4. ${}^{33}{C}_3-{(}^{10}{C}_3+{}^{10}{C}_4+{}^{12}{C}_3)$

Q. The angle between pair of lines joining $(0,0)$ to the points of intersection of the curve $x^2+y^2=9$ with the lines $x+y=3$ is $\frac{\pi }{k}$, then $k=$

Q. A chord $ax+y=1$ subtends ${90}^{\circ }$ at the center of the circle $x^2+y^2=\frac{3}{2}.$ Then the value of $a$ is

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

Q. If the equation $6x^2-\alpha xy-3y^2-24x+3y+\beta =0$ represents a pair of straight lines that intersect on the $x-$axis, then the value of $20\alpha -\beta$ is

Q.

A pair of perpendicular straight lines passes through the origin and also through the point of intersection of the curve $x^2+y^2$ =4 with x+y=a. The set containg the value of 'a' is

1. {−2, 2 }

2. {−3, 3 }

3. {−4, 4 }

4. {−5, 5 }

Q. One bisector of the angle between the lines given by $a(x-1{)}^2+2h(x-1)y+b{y}^2=0$ is $2x+y-2=0$, then

1. 
2. 
3. 
4. 

Q. Prove that the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}$ intersects and find the point of intersection.

Q. Evaluate $f\left(x\right)=\sqrt{{\log }_{0.5}(3x-8)-{\log }_{0.5}(x^2+4)}$?

Q. If the equation $6x^2-\alpha xy-3y^2-24x+3y+\beta =0$ represents a pair of straight lines that intersect on the $x-$axis, then the value of $20\alpha -\beta$ is

Q. A pair of perpendicular straight lines passing through the origin also passes through the points of intersection of the curve $x^2+y^2=4$ with the line $x+y=a$, then value(s) of $a$ can be

1. $2$
2. $3$
3. $-2$
4. $-3$

Q.

What's the equation of the line joining 2 points with eccentric angles ${30}^{\circ }$ and ${60}^{\circ }$ in the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$?

1. 

2. x~-~3y~-~15{\sqrt{3}}~+~8~=~0\)

3. (b)

4. x~+~3y~+~15{\sqrt{3}}~-~8~=~0\)

Q.

A chord $ax+y=1$ subtends ${90}^{\circ }$ at the centre of the circle $x^2+y^2=\frac{3}{2}.$ Then the value of $a$ is

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

Q. The equation of the lines represented by $4x^2+24xy+11y^2=0$ is/are

1. $2x+y=0$
2. $2x+11y=0$
3. $x-y=0$
4. $y-2x=0$