Foot of Perpendicular from a Point on a Line

Q. A perpendicular is drawn from a point on the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1}$ to the plane $x+y+z=3$ such that the foot of the perpendicular $Q$ also lies on the plane $x-y+z=3$. Then the co-ordinates of $Q$ are :

1. $(2,0,1)$
2. $(-1,0,4)$
3. $(4,0,-1)$
4. $(1,0,2)$

Q. Let the foot of perpendicular from a point $P(1,2,–1)$ to the straight line $L:\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ be $N$. Let a line be drawn from $P$ parallel to the plane $x+y+2z=0$ which meets $L$ at point $Q$. If $\alpha$ is the acute angle between the lines $PN$ and $PQ,$ then $\cos \alpha$ is equal to

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

Q.

The equations of the perpendicular bisectors of the sides $AB$ and $AC$ of a triangle $ABC$ are $x-y+5=0$ and $x+2y=0$ respectively. If the co-ordinates of $A$ are $(1,-2)$, then the equation of $BC$ is

1. $23x+14y-40=0$
2. $14x+23y-40=0$
3. $23x-14y+40=0$
4. None of these

Q.

The coordinates of the foot of the perpendicular from the point (2, 3) on the line $x+y-11=0$ are

1. (-6, 5)

2. (5, 6)

3. (-5, 6)

4. (6, 5)

Q. If the equation of the curve on reflection of the ellipse $\frac{(x-4{)}^2}{16}+\frac{(y-3{)}^2}{9}=1$ about the line $x-y-2=0$ is $16x^2+9y^2+k_{1}x-36y+k_2=0$, then $\frac{k_1+k_2}{22}$ is

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

Q. The line $\frac{x}{a}+\frac{y}{b}=1$ moves in such a way that $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$ where $c$ is a constant. The locus of foot of perpendicular from the origin on the given line will be

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

Q. The image of the point (4, -3) with respect to the line y = x is

1. (–4, –3)
2. (3, 4)
3. (–4, 3)
4. (–3, 4)

Q. Consider a triangle having vertices $A(–2,3),B(1,9)$ and $C(3,8)$. If a line $L$ passing through the circumcenter of triangle $ABC$, bisects line $BC$, and intersects $y$-axis at point $\left(0.\frac{\alpha }{2}\right),$ then the value of real number $\alpha$ is

Q.

The equation $y^2+4x+4y+k=0$ represents a parabola whose latus rectum is

1. $1$

2. $2$

3. $3$

4. $4$

Q. The coordinates of the foot of perpendicular drawn from the point $(1,-2)$ on the line $y=2x+1$, is

1. $(-1,-1)$
2. $(1,1)$
3. $(1,-1)$
4. $(-1,1)$

Q.

A line forms a triangle of area $54\sqrt{3}$ square units with the coordinate axes. Find the equation of the line if the perpendicular drawn from the origin to the line makes an angle of $60º$ with the$x-$axis

Q. The orthocentre of the triangle PAB is

1. $(\frac{7}{5},\frac{25}{8})$
2. $(\frac{8}{25},\frac{7}{5})$
3. $(\frac{11}{5},\frac{8}{5})$
4. $(5,\frac{8}{7})$

Q.

Find the coordinates of the foot of the perpendicular from the point $(-1,3)$ to the line $3x-4y-16=0.$

Q. The area of the quadrilateral formed by the lines $4x-3y-a=0,3x-4y+a=0$, $4x-3y-3a=0$ and $3x-4y+2a=0$ is

1. $\frac{2a^2}{11}\text{sq. units}$
2. $\frac{2a^2}{9}\text{sq. units}$
3. $\frac{a^2}{7}\text{sq. units}$
4. $\frac{2a^2}{7}\text{sq. units}$

Q.

Find the projection of the point $(1,0)$ on the line joining the points $(-1,2)$ and $(5,4).$

Q. 19. The projection of the line segment joining the points(1, 2, 3) and (4, 5, 6) on the plane 2x +y+z=1and also find the projection of these points on the line x-1/2=y-3/2=z+4/1

Q. The coordinates of the point on the parabola $y=x^2+7x+2,$ which is nearest to the straight line y = 3x - 3 are

1. (-2, -8)
2. (2, 20)
3. (-1, -4)
4. (1, 10)

Q.

Find the direction cosines of the line segment joining the two points (1, - 2, 3). and ( 2, 4, 5)

1. 

2. 

3. 

4. 

Q. A variable line $\frac{x}{a}+\frac{y}{b}=1$ moves such that, the harmonic mean of $a^2$ and $b^2$ is $32.$ If the locus of the foot of the perpendicular from the origin to the above line is a circle with radius ${}^{\prime }{r}^{\prime },$ then the value of $4r^2+1$ is

Q. If the coordinates of the foot of the perpendicular drawn from the point $(1,-2)$ on the line $y=2x+1$ is $(\alpha ,\beta )$, then the value of $|\alpha +\beta |$ is

Q. A line parallel to y axis then X2-X1=0 ..why???

Q. A straight line passes through a fixed point $(h,k)$. The locus of the foot of perpendicular on it drawn from the origin is

1. $x^2+y^2=hx-ky$
2. $x^2+y^2=kx-hy$
3. $x^2+y^2-hx-ky=0$
4. $x^2+y^2-kx-hy=0$

Q. The straight lines $x+y=0,3x+y-4=0,x+3y-4=0$ form a triangle which is

1. isosceles
2. equilateral
3. right angled
4. none of these

Q. find the image of the point (2, 3, -1) in the point (3, 0, 4)

Q. If a $△ABC$ has vertices $A(-1,7),B(-7,1)$ and $C(5,-5)$, then its orthocentre has coordinates:

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

Q. Find the coordinates of the point which divides the line segment joining $(3,4)$ and $(-6,2)$ in the ratio $3:2$ externally.

Q. For non-negative real numbers $h_1,h_2,h_3,k_1,k_2,k_3,$ if the algebraic sum of the perpendiculars drawn from the points $(2,k_1),$ $(3,k_2),$ $(7,k_3),$ $(h_1,4),$ $(h_2,5),$ $(h_3,-3)$ on a variable line passing through $(2,1)$ is zero, then

1. $h_1=1,h_2=1,h_3=2$
2. $k_1=k_2=k_3=0$
3. $h_1=h_2=h_3=0$
4. $k_1=1,k_2=2,k_3=3$

Q. The equations of the perpendicular bisectors of the sides $AB$ and $AC$ of a triangle $ABC$ are $x-y+5=0$ and $x+2y=0$ respectively. If the co-ordinates of $A$ are $(1,-2)$, then the equation of $BC$ is

1. $23x+14y-40=0$
2. $14x+23y-40=0$
3. $23x-14y+40=0$
4. None of these

Q. The point of the form $(a,-a)$ always lies on the line

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