Equation of Perpendicular from a Point on a Line

Q.

If the line $\frac{x}{a}+\frac{y}{b}=1$ passes through the points $(2,-3)$ and $(4,-5)$, then $(a,b)=$

1. $(1,1)$

2. $(-1,1)$

3. $(1,-1)$

4. $(-1,-1)$

Q. If the length of the perpendicular from the point $(\beta ,0,\beta )(\beta \ne 0)$ to the line, $\frac{x}{1}=\frac{y-1}{0}=\frac{z+1}{-1}$ is $\sqrt{\frac{3}{2}}$, then $\beta$ is equal to :

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

Q.

The equations of the lines represented by the equation $a{x}^2+\left(a+b\right)xy+b{y}^2+x+y=0$ are

1. $ax+by+1=0,x+y=0$

2. $ax+by-1=0,x+y=0$

3. $ax+by+1=0,x-y=0$

4. None of these

Q.

The differential equation of all parabolas whose axes are parallel to y-axis, is

1. $\raisebox{1ex}{${\mathrm{d}}^3\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dx}}^3=0$}\right.$

2. $\raisebox{1ex}{${\mathrm{d}}^2\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dy}}^2=\mathrm{c}$}\right.$

3. $\left(\raisebox{1ex}{${\mathrm{d}}^3\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dx}}^3$}\right.\right)+\left(\raisebox{1ex}{${\mathrm{d}}^2\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dy}}^2$}\right.\right)=0$

4. $\left(\raisebox{1ex}{${\mathrm{d}}^2\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dy}}^2$}\right.\right)+2\left(\raisebox{1ex}{$\mathrm{dy}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dx}$}\right.\right)=\mathrm{c}$

Q.

The point $\mathrm{P}(\mathrm{a},\mathrm{b})$ lies on the straight line $3x+2y=13$ and the point $Q(b,a)$ lies on the straight line $4x-y=5$, then the equation of line $\mathrm{PQ}$ is

1. $x-y=5$

2. $x+y=5$

3. $x+y=-5$

4. $x-y=-5$

Q.

Prove that the line through A(0, -1, -1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(-4, 4, 4).

Q.

Maximize Z = - x + 2y, subject to the constraints are x $\ge$ 3, x + y $\ge$ 5, x + 2y $\ge$ 6 and x, y $\ge$ 0.

Q.

Two lines $\frac{[x-1]}{2}=\frac{[y+1]}{3}=\frac{[z-1]}{4}$ and $\frac{[x-3]}{1}=\frac{[y-k]}{2}=\frac{z}{1}$ intersect at a point, if$k=$

1. $\frac{2}{9}$

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

3. $\frac{9}{2}$

4. $\frac{1}{6}$

Q. The angle between the lines joining the points $(1,1,0),(-3,\sqrt{3}+1,3)\text{and}(0,-1,0),(-1,\sqrt{3}-1,\lambda )$ is ${\cos }^{-1}\left(\sqrt{\frac{7}{16}}\right)$.If $\lambda$ is an integer then $\lambda$ is:

Q.

The differential equation of the rectangular hyperbola, where axes are the asymptotes of the hyperbola, is

1. $\mathrm{y}\frac{d\mathrm{y}}{d\mathrm{x}}=\mathrm{x}$

2. $\mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}}=-\mathrm{y}$

3. $\mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}}=\mathrm{y}$

4. $\mathrm{x}\mathrm{dy}+\mathrm{y}\mathrm{dx}=\mathrm{c}$

Q. The vector and the cartesian equations of the line through the point $(5,–2,4)$ and which is parallel to the vector $2\hat{i}-\hat{j}+3\hat{k}$ are __________

Q.

Find the equation of the line passing through the point of intersection of$7x+6y=71$ and $5x-8y=-23$ and perpendicular to the line$4x-2y=1.$

Q. If from the point P(a, b, c) perpendicular PL, PM be drawn to YOZ and ZOX planes, then the equation of the plane OLM is

1. $\frac{x}{a}+\frac{y}{b}-\frac{z}{c}=0$
2. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0$
3. $\frac{x}{a}-\frac{y}{b}+\frac{z}{c}=0$
4. $\frac{x}{a}-\frac{y}{b}-\frac{z}{c}=0$

Q. If O is the origin and the coordinates of P is $(1,2,-3)$, then find the equation of the plane passing through P and perpendicular to OP.

1. $x-2y-3z=-15$
2. $x+2y-3z=14$
3. $x-2y+3z=15$
4. $x-2y-3z=15$

Q. The mirror image of the point $(1,2,3)$ in a plane is $(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3})$. Which of the following points lies on this plane?

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

Q. The cartesian equation of a line is $\frac{x-4}{3}=\frac{y+1}{-2}=\frac{z+2}{5}$. Find the vector equation of line.

Q. The locus of the mid-points of the perpendiculars drawn from points on the line, $x=2y$ to the line $x=y$ is :

1. $2x-3y=0$
2. $3x-2y=0$
3. $5x-7y=0$
4. $7x-5y=0$

Q.

The number of integer values of $m$, for which the $x-$coordinate of the point of intersection of the lines $3x+4y=9$ and $y=mx+1$ is also an integer, is

1. $2$

2. $0$

3. $4$

4. $1$

Q.

If the foot of the perpendicular of a point $P(2,-3,1)$ with respect to line L is $(\frac{-22}{7},\frac{-3}{14},\frac{-13}{14})$, then the coordinates of it's image $I$ is .

1. $\frac{-58}{7},\frac{36}{14},\frac{-40}{14}$
2. $\frac{-44}{7},\frac{-6}{14},\frac{-26}{14}$
3. $\frac{44}{7},\frac{6}{14},\frac{-30}{14}$
4. $\frac{-44}{7},\frac{6}{14},\frac{-30}{14}$

Q. The foot of the perpendicular of P(2, -3, 1) on the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-2}{-1}$ is Q. If the DRs of the line segment joining P and Q is l, m, n, then .

1. 2l+3m+n = 1
2. 2l+3m-n = 0
3. $\frac{l}{2}=\frac{m}{3}=\frac{n}{-1}$
4. $\frac{l}{2}=\frac{m}{3}=\frac{n}{1}$

Q. Find the equation of plane passing through the point $(3,2,4)$ and perpendicular to the $X-$axis.

Q.

During the summer Abrielle is a cashier at a grocery store and Zane serves ice cream to show their earnings Abrielle makes a graph and Zane makes a table.

Which statement is true.

Abrielles slope is $10.5$

Zane

$2\mathrm{hr}$	$\$18.50$
$5\mathrm{hr}$	$\$46.25$
$7\mathrm{hr}$	$\$64.75$

1. Zane earns $\$8$ more than Abrielle

2. Zane earns $\$4.75$ more than Abrielle

3. Abrielle earns $\$1.25$ more than Zane

4. Abrielle earns$\$2.50$ more Zane

Q. If the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar, then $k$ can have

1. exactly one value
2. exactly two values
3. exactly three values
4. any value

Q.

The number of integral values of m, for which the x-coordinate of the point of intersection of the lines$3x+4y=9$ and $y=mx+1$ is also an integer is (1) (2)(3) (4)

1. $0$

2. $2$

3. $4$

4. $1$

Q. The equation of the plane passing through $(2,-3,1)$ and is normal to the line joining the points $(3,4,-1)$ and $(2,-1,5)$ is given by

1. $x+5y-6z+19=0$
2. $x-5y+6z-19=0$
3. $x+5y+6z+19=0$
4. $x-5y-6z-19=0$

Q.

If the lines $3\left(\alpha -1\right)y-6x=2$ and $4y-8x+10=0$ are parallel, then find the value of $\alpha$.

Q. Find the equations of the plane through the point $(x_1,y_1,z_1)$ and perpendicular to the straight line $\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$

1. $l(x-x_1)+m(y-y_1)+n(z-z_1)=0.$
2. $l(x-x_1)-m(y-y_1)+n(z-z_1)=0.$
3. $l(x-x_1)+m(y-y_1)-n(z-z_1)=0.$
4. none of these

Q. The distance between the line $\overline{r}=2\hat{i}-2\hat{j}-3\hat{k}+\lambda (\hat{i}-\hat{j}+4\hat{k})$
and the plane $\overline{r}.(\hat{i}+5\hat{j}+\hat{k})=5$

1. $\frac{10}{3\sqrt{3}}$
2. $\frac{10}{3}$
3. $\frac{10}{9}$
4. $\frac{1}{9}$

Q. The equation of plane passing through a point $A(2,-1,3)$ and parallel to the vectors $a=(3,0,-1)$ and $b=(-3,2,2)$ is:

1. $2x-3y+6z+25=0$
2. $3x-2y+6z-25=0$
3. $3x-2y+6z+25=0$
4. $2x-3y+6z-25=0$

Q. Find the equation of the plane through the point $(1,1,1)$ and perpendicular to the line $x-2y+z=2,4x+3y-z+1=0$