Condition For Perpendicularity

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

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

Q. Equation of a tangent to the hyperbola $5x^2-y^2=5$ and which passes through an external point $(2,8)$ is

1. $3x-y+2=0$
2. $3x+y-14=0$
3. $23x-3y-22=0$
4. $3x-23y+178=0$

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 $(0,4)$ and $(0,2)$ are respectively the vertex and focus of a parabola, then its equation is

1. $x^2+8y=32$
2. $y^2+8x=32$
3. $y^2-8x=32$
4. $x^2-8y=32$

Q.

Which of the following statements are correct for oblique hyperbola $xy=8$
1. Equation of tangent at P(4, 2) is $x+2y=8$

2. Equation of normal at P(t) is $x{t}^3-yt=2\sqrt{2}(t4-1)$

1. only 1

2. only 2

3. Both 1 & 2

4. None of these

Q. One end of a diameter of the circle $x^2+y^2-6x+5y-7=0$ is (-1, 3). Find the co-ordinates of the other end.

Q. The centre of a clock is taken as origin At 4.30 pm the equation of line along minute hand is x = 0 Therefore at this instant the equation of line along the hour hand will be

1. $x-y=0$
2. $x+y=0$
3. $y=\sqrt{2x}$
4. $y=\frac{x}{\sqrt{2}}$

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. A ray of light travels along the line $2x-3y+5=0$ and strikes a plane mirror lying along the line $x+y=2$. The equation of the straight line containing the reflected ray is

1. $2x-3y+3=0$
2. $21x-7y+1=0$
3. $21x+7y-1=0$
4. $3x-2y+3=0$

Q. The line $y=-1$ intersects the parabola $y=x^2-10x+24$ at one point. Find the coordinates of the point of the intersection.

1. $(-5,-1)$
2. $(24,-1)$
3. $(0,-1)$
4. $(5,-1)$
5. $(35,-1)$

Q. $L_1\&L_2$ are the two lines which passes through the point $(1,-1)$ and tangent to the curve $y=x^2-3x+2.$ Then which of the following point(s) lie on the hyperbola having $L_1\&L_2$ as its asymptotes and passing through the origin?

1. $(2,-2)$
2. $(\frac{-2}{3},-2)$
3. $(\frac{8}{3},0)$
4. $(-2,-2)$

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. A ray of light is coming along the line $y=4$ from the positive direction of $x-axis$ & strikes a concave mirror whose intersection with the plane $XOY$ is a parabola $y^2=16x$, then equation of reflected ray is

1. $3x+2y=16$
2. $4x+3y=16$
3. $5x+4y=9$
4. None of these

Q. A light beam emanating from the point A$(3,10)$ reflects from the line $2x+y-6=0$ and then passes through the point B$(5,6)$. The equation of the incident and reflected beams respectively are:

1. $4x-3y+18=0andy=6$
2. $x-2y+8=0andx=5$
3. $x+2y-8=0andy=6$
4. None of these

Q. If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord to the circle with centre $(2,1)$, then the radius of the circle is:

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

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 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 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. Assertion :A ray of light passes through $(0,0)$ after reflection at the point $P(x,y)$ of any curve becomes parallel to x-axis, the equation of the curve may be a parabola $y^2=2x+1$ Reason: A ray of light parallel to axis after reflection from parabolic mirror always passes through $(0,0)$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. The radius of the locus by the point represented by $z$, when $arg\frac{z-1}{z+1}=\frac{\pi }{4}$, is

1. $\sqrt{2}\pi$
2. $\frac{\pi }{\sqrt{2}}$
3. $noneofthese$
4. $\sqrt{2}$

Q. The area of the closed figure bounded by $y=x,y=-x$ & the tangent to the curve $y=\sqrt{x^2-5}$ at the point (3, 2) is

1. $5$
2. $2\sqrt{5}$
3. $10$
4. $\frac{5}{2}$

Q. A ray of light is coming along the line $y=b$ from the positive direction of $x-$ axis and strikes the concave mirror whose interaction with $x-$ plane is a parabola $y^2=4ax$. Find the equation of the reflected ray. Assume $a,b$ are positive.

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

Q. ABC is a variable triangle such that A is $(1,2)$ and B and C on the line $y=x+\lambda$ . Then the locus of the orthocenter of $\Delta ABC$ is

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

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. $(1,0)$