Condition For Perpendicularity

Q.

If the equation $a{x}^2+2hxy+b{y}^2=0$ has the one line as the bisector of angle between the coordinates axes, then

1. ${(a–b)}^2=h^2$

2. ${(a+b)}^2=h^2$

3. ${(a–b)}^2=4h^2$

4. ${(a+b)}^2=4h^2$

Q. If the line $x+y=1$ touches the parabola $y^2-y+x=0$, then the coordinates of the point of contact are

1. $(1,1)$
2. $(\frac{1}{2},\frac{1}{2})$
3. $(0,1)$
4. $(1,0)$

Q. If at $x=1,y=2x$ is tangent to the parabola $y=a{x}^2+bx+c,$ then which of the following option(s) is/are correct?

1. $a=c$
2. $a,b,c$ are in A.P
3. infinite such parabolas exist
4. $2a+c=2$

Q. If the line $x+y=1$ touches the parabola $y^2-y+x=0$, then the coordinates of the point of contact are

1. $(1,1)$
2. $(\frac{1}{2},\frac{1}{2})$
3. $(0,1)$
4. $(1,0)$

Q. If a ray of light along the line $y=4$ gets reflected from a parabolic mirror whose equation is $(y-2{)}^2=4(x+1)$, then equation of reflected ray will be

1. $y=0$
2. $x=0$
3. $y=2$
4. $x=2$

Q. If a ray of light along the line $y=4$ gets reflected from a parabolic mirror whose equation is $(y-2{)}^2=4(x+1)$, then equation of reflected ray will be

1. $y=0$
2. $x=0$
3. $y=2$
4. $x=2$

Q. If from a point $P$, $3$ normals are drawn to parabola $y^2=4ax$, then the locus of $P$ such that one of the normal is angular bisector of other two normals is

1. $(2x-a)(x-5a{)}^2=27a{y}^2$
2. $(2x-a)(x+5a{)}^2=27a{y}^2$
3. $(2x-a)(x-5a)=27a{y}^2$
4. $(2x-a)(x+5a)=27a{y}^2$

Q. If from a point $P$, $3$ normals are drawn to parabola $y^2=4ax$, then the locus of $P$ such that one of the normal is angular bisector of other two normals is

1. $(2x-a)(x-5a{)}^2=27a{y}^2$
2. $(2x-a)(x+5a{)}^2=27a{y}^2$
3. $(2x-a)(x-5a)=27a{y}^2$
4. $(2x-a)(x+5a)=27a{y}^2$