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.

The equation of pair of straight lines joining the point of intersection of the curve $x^2+y^2=4$ and y - x = 2 to the origin, is

1. $x^2+y^2=(y-x{)}^2$

2. $x^2+y^2+(y-x{)}^2=0$

3. $x^2+y^2=4(y-x{)}^2$

4. $x^2+y^2+4(y-x{)}^2=0$

Q. If the curve $x^2+2y^2=2$ intersects the line $x+y=1$ at two points $P$ and $Q,$ then the angle subtended by the line segment $PQ$ at the origin is:

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

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 pair of straight lines joining the point of intersection of the curve $x^2+y^2=4$ and y - x = 2 to the origin, is

1. $x^2+y^2=(y-x{)}^2$

2. $x^2+y^2+(y-x{)}^2=0$

3. $x^2+y^2=4(y-x{)}^2$

4. $x^2+y^2+4(y-x{)}^2=0$

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. 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. 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. 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. 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. 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. $-3$