Perpendicular Distance of a Point from a Line

Q.

A person standing at the junction of two straight paths represented by the equations $2x-3y+4=0$ and $3x+4y-5=0$ wants to reach the path whose equation is $6x-7y+8=0$ in the least time. Find the equation of the path that he should follow.

Q. The equation of a tangent to the parabola $y^2=8x$ which makes an angle ${45}^{\circ }$ with the line $y=3x+5$ is

1. $2x+y+1=0$
2. $y=2x+1$
3. $x+2y+8=0$
4. $x-2y+8=0$

Q.

The equation of tangent at $(–4,–4)$ on the curve $x^2=–4y$ is

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

2. $2x–y–6=0$

3. $2x+y–4=0$

4. $2x–y+4=0$

Q. The distance between the lines $3x-4y+9=0$ and $6x-8y-15=0$ is ___ units

1. $\frac{20}{33}$
2. $\frac{10}{33}$
3. $\frac{33}{10}$
4. $\frac{33}{20}$

Q.

If the coordinates of a variable point $P$ be $\left(a\cos \left(\theta \right),b\sin \left(\theta \right)\right)$ where $\theta$ is a variable quantity , find the locus of $P$

Q. The set of points on the axis of the parabola $y^2-2y-4x+5=0$ from which all the three normals to the parabola are real is :

1. $\left\{\right(x,1);x>3\}$
2. $\left\{\right(x,1);x\ge 3\}$
3. $\left\{\right(x,3);x\ge 1\}$
4. $\left\{\right(x,-3);x>3\}$

Q. The locus of the point of intersection of perpendicular tangent drawn to each one of the parabola $y^2=4x+4$ and $y^2=8x+16$ is

1. $x=-3$
2. $x=-12$
3. $x=-8$
4. $x=-4$

Q. The condition that the straight line $lx+my+n=0$ touches the parabola $x^2=4ay$ is

1. $bn=a{m}^2$
2. $a{l}^2-mn=0$
3. $ln=a{m}^2$
4. $am=l{n}^2$

Q. Let $\sqrt{3}\hat{i}+\hat{j},\hat{i}+\sqrt{3}\hat{j}$ and $\beta \hat{i}+(1-\beta )\hat{j}$ respectively be the position vectors of the points $A,B$ and $C$ with respect to the origin $O$. If the distance of $C$ from the bisector of the acute angle between $OA$ and $OB$ is $\frac{3}{\sqrt{2}}$, then the sum of all possible values of $\beta$ is :

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

Q. Equation(s) of the line perpendicular to $x=y$ and at a distance of $1$ unit from origin is/are

1. $x+y+\sqrt{2}=0$
2. $3x+4y+5=0$
3. $x+y-\sqrt{2}=0$
4. $3x+4y-5=0$

Q. If the line $y-\sqrt{3}x+3=0$ cuts the curve $y^2=x+2$ at $A$ and $B,P$ is a point on the line whose ordinate is $0.$ Then $|PA.PB|=$

1. $\frac{4}{3}(\sqrt{3}+2)$
2. $\frac{4}{3}(2-\sqrt{3})$
3. $\frac{2}{3}(\sqrt{3}+2)$
4. $\frac{2}{3}(2-\sqrt{3})$

Q. The number of mutually perpendicular tangents that can be drawn from the curve $y=||1-e^x|-2|$ to the parabola $x^2=-4y$ is

Q. The equation of perpendicular bisectors of sides $AB,BC$ of $\Delta ABC$ are $x-y-5=0$ and $x+2y=0$ respectively. If $A\equiv (1,-2),C\equiv (\alpha ,\beta )$, then $\alpha +\beta$ is equal to

Q. Let $L$ be a line obtained from the intersection of two planes $x+2y+z=6$ and $y+2z=4$. If point $P(\alpha ,\beta ,\gamma )$ is the foot of perpendicular from $(3,2,1)$ on $L$, then the value of $21(\alpha +\beta +\gamma )$ equals:

1. $102$
2. $68$
3. $142$
4. $136$

Q. The equation of tangent to the parabola $y^2=6x$ at point $(6,6)$ is

1. $2x+3y-30=0$
2. $2x-y-6=0$
3. $2y-x-6=0$
4. $x+y-12=0$

Q. If the foot of the perpendicular from point $(4,3,8)$ on the line $L_1:\frac{x-a}{l}=\frac{y-2}{3}=\frac{z-b}{4},$
$l\ne 0$ is $(3,5,7),$ then the shortest distance between the line $L_1$ and line $L_2:\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ is equal to :

1. $\frac{1}{\sqrt{3}}$
2. $\frac{1}{\sqrt{6}}$
3. $\sqrt{\frac{2}{3}}$
4. $\frac{1}{2}$

Q. The point of intersection of two tangents at the ends of the latus rectum to the parabola $(y+3{)}^2=8(x-2)$ is

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

Q.

The coordinates of a point at unit distance from the lines 3x - 4y + 1 = 0 and 3x + 6y + 1 = 0 are

1. $(\frac{6}{5},\frac{-1}{10})$

2. $(0,\frac{-1}{10})$

3. $(\frac{-2}{5},\frac{-13}{10})$

4. $(\frac{-8}{5},\frac{3}{10})$

Q. One vertex of a square $ABCD$ is $A(-1,1)$ and the equation of diagonal $BD$ is $3x+y-8=0.$ Then

1. $C\equiv (-7,-1)$
2. $C\equiv (5,3)$
3. $\text{Area of square}=40\text{sq. units}$
4. $\text{Area of square}=20\text{sq. units}$

Q. The length of perpendicular from the focus $S$ of the parabola $y^2=4ax$ on the tangent at any point $P$ on the parabola is
($O$ is the origin)

1. $\sqrt{OS\cdot SP}$
2. $OS\cdot SP$
3. $OS+SP$
4. $\frac{OS}{OP}$

Q. The equation of circle which passes through focus of parabola $x^2=4y$ and touches it at $(6,9)$ is

1. $x^2+y^2+18x-28y+27=0$
2. $x^2+y^2+24x-27y+26=0$
3. $x^2+y^2+48x-12y+11=0$
4. $x^2+y^2+18x-22y+21=0$

Q. In $\overline{r}.\overline{n}=d,|d|$ represents the distance between

1. Origin and any point on the plane
2. Points where the plane intersects the axis
3. Foot of perpendicular along $\hat{n}$ and origin
4. None of these

Q. One vertex of a square $ABCD$ is $A(-1,1)$ and the equation of diagonal $BD$ is $3x+y-8=0.$ Then

1. $C\equiv (-7,-1)$
2. $C\equiv (5,3)$
3. $\text{Area of square}=40\text{sq. units}$
4. $\text{Area of square}=20\text{sq. units}$

Q. If two tangents to $y^2=4ax$ make angles ${\theta }_1,{\theta }_2$ with positive $x-$ axis such that $\cos {\theta }_1\cdot \cos {\theta }_2=k,$ then the locus of their point of intersection is

1. $x^2=k^2\left[\right(x-a{)}^2+y^2]$
2. $x^2=k^2\left[\right(x+a{)}^2+y^2]$
3. $x^2=k^2\left[\right(x-a{)}^2-y^2]$
4. $x^2=\left[\right(x+a{)}^2+y^2]$

Q. A line passing through $A(1,2)$ is perpendicular to the lines $3x+4y-2=0$ and $3x+4y+7=0$ and intersecting them respectively at the points $P$ and $Q,$ then $\frac{AQ}{AP}$ is equal to

Q. The point of intersection of tangents at the points on the parabola $y^2=4x$ whose ordinates are $4$ and $6$ is

1. $(6,5)$
2. $(7,3)$
3. $(9,10)$
4. $(6,10)$

Q. A variable line passes through a fixed point P, The algebraic sum of the perpendicular distances from (2, 0), (0, 2) and (1, 1) to the line is zero, then the coordinates of the P are

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

Q. If variable parameter $\alpha ,\beta ,\gamma \in R$ and satisfy the relations ${\alpha }^2+2{\beta }^2+3\alpha =1+{\gamma }^2$ and $2{\alpha }^2+4{\beta }^2=2{\gamma }^2+5\beta ,$ then

1. Minimum value of ${\alpha }^2+{\beta }^2$ is $\frac{4}{61}$
2. Minimum value of ${\alpha }^2+{\beta }^2$ is $\frac{2}{17}$
3. Minimum value of ${\alpha }^2+{\beta }^2-2\alpha +1$ is $\frac{16}{61}$
4. Minimum value of ${\alpha }^2+{\beta }^2-2\alpha +1$ is $\frac{1}{17}$

Q. The equation of tangent to the parabola $y^2=6x$ at point $(6,6)$ is

1. $2x+3y-30=0$
2. $2x-y-6=0$
3. $2y-x-6=0$
4. $x+y-12=0$

Q. Equation(s) of the common tangent to the parabola $y^2=24x$ and the circle $x^2+y^2=18$ is/are

1. $2x+y+3=0$
2. $x-y+6=0$
3. $x+y+6=0$
4. $x-2y+8=0$