Parametric Form of a Straight Line

Q. A variable line $L$ is drawn through $O(0,0)$ to meet the lines $L_1:x+2y-3=0$ and $L_2:x+2y+4=0$ at points $M$ and $N$ respectively. A point $P$ is taken on line $L$ such that $\frac{1}{O{P}^2}=\frac{1}{O{M}^2}+\frac{1}{O{N}^2}$. Then the locus of $P$ is

1. $x^2+4y^2=\frac{144}{25}$
2. $(x+2y{)}^2=\frac{144}{25}$
3. $4x^2+y^2=\frac{144}{25}$
4. $(x-2y{)}^2=\frac{144}{25}$

Q. Parametric form of a straight line passing through $(4,5)$ and making an angle ${60}^{\circ }$ with $x$-axis in the positive direction is
(where $\lambda$ be any parameter)

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

Q. The parametric equation of the line is given by
$x=-2+\frac{r}{\sqrt{10}};y=1+\frac{3r}{\sqrt{10}}$, then for the line

1. $x$ intercept is $\frac{7}{3}$
2. $y$ intercept is $7$
3. slope is $3$
4. $x$ intercept is $-\frac{7}{3}$

Q. The slope of a straight line passing through $A(-2,3)$ is $-\frac{4}{3}$. The point(s) on the line that are $10$ units away from $A$ is/are

1. $(8,0)$
2. $(-8,11)$
3. $(4,-5)$
4. $(4,11)$

Q.

If one vertex of an equilateral triangle of side $a$ lies at the origin and the other lies on line $x-\sqrt{3}y=0$, then the co-ordinates of the third vertex are

Q. If the straight line drawn through the point $P(\sqrt{3},2)$ and inclined at an angle of $\frac{\pi }{6}$ with the positive direction of $x-$axis meets the line $\sqrt{3}x-4y+8=0$ at point $Q,$ then the length of $PQ$ is

Q. If the straight line through the point $P(3,4)$ makes an angle $\frac{\pi }{6}$ with the $x$-axis and meets the line $3x+5y+1=0$ at $Q,$ then length of $PQ$ is

1. $30(3\sqrt{3}-5)$ units
2. $30(\sqrt{3}-1)$ units
3. $\sqrt{3}-1$ units
4. $30$ units

Q. The parallelism condition for two straight lines one of which is specified by the equation ax+by+c=0 the other being represented parametrically by $x=\alpha t+\beta ,y=\gamma t+\delta$ is given by

1. $\alpha \gamma -b\alpha =0,\beta =\delta =c=0$
2. $a\alpha -b\gamma =0,\beta =\delta =0$
3. $a\alpha +b\gamma =0$
4. $a\gamma =b\alpha =0$

Q.

Find the area of the region bounded by the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1.$

Q.

Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to the line 3x - 4y + 8 = 0.

Q.

Find the distance of the line 2x + y = 3 from the point (-1, -3) in the direction of the line whose slope is 1.

Q.

A line passes through a point A (1, 2) and makes an makes an angle of ${60}^{\circ }$ with the x-axis and intersects the line x + y = 6 at the point P. Find AP.

Q. A straight line $L$ is drawn through the point $A(2,1)$ such that its point of intersection with $x+y=9$ is at a distance of $3\sqrt{2}$ unit's from $A$. Then

1. Inclination of $L$ is $\frac{\pi }{4}$
2. Inclination of $L$ is $\frac{\pi }{6}$
3. $L$ cuts the $x-$axis at $(1,0)$
4. $L$ passes through $(5,4)$

Q. A line which passes through $P(4,5)$ and making an angle of ${30}^{\circ }$ with positive direction of $x-$axis. Then coordinates of point which is at a distance $4$ unit's from the line on either side of $P$, is

1. $(4+2\sqrt{3},7),(4-2\sqrt{3},3)$
2. $(4-2\sqrt{3},7),(4+2\sqrt{3},3)$
3. $(4-2\sqrt{3},7),(4+2\sqrt{3},7)$
4. $(4+2\sqrt{3},3),(4+2\sqrt{3},7)$

Q. Let $S=(0,2\pi )-\{\frac{\pi }{2},\frac{3\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}\}$. Let $y=y\left(x\right),x\in S$ be the solution curve of the differential equation $\frac{dy}{dx}=\frac{1}{1+\sin 2x},y\left(\frac{\pi }{4}\right)=\frac{1}{2}$. If the sum of abscissas of all the points of intersection of the curve $y=y\left(x\right)$ with the curve $y=\sqrt{2}\sin x\text{is}\frac{k\pi }{12}$, then $k$ is equal to

Q.

A straight line drawn through the point A (2, 1) making an angle $\pi /4$ with positive x-axis intersects another line x + 2y + 1 = 0 in the point B. Find length AB.

Q. The equation of the medians of a triangle formed by the lines $x+y-6=0,x-3y-2=0$ and $5x-3y+2=0$, is

1. $x=2,x-9y-14=0$ and $7x-9y+2=0$
2. $x=2,x+9y-14=0$ and $7x-9y-2=0$
3. $x=2,x-9y-14=0$ and $7x+9y+2=0$
4. $x=2,x+9y+14=0$ and $x-9y-2=0$

Q.

Find the equation of straight line passing through (-2, -7) and having an intercept of length 3 between the straight lines 4x + 3y = 12 and 4x + 3y = 3.

Q. The slope of the straight line passing through A(3, 2) is $\frac{3}{4}$. The coordinates of the points that are 5 units away from A are:

1. (7, 5)
2. (-1, -1)
3. (-7, -5)
4. (1, 1)

Q.

A line a drawn through A (4, -1) parallel to the line 3 x - 4 y + 1 = 0. Find the coordinates of the two points on this line which are at a distance of 5 units from A.

Q. The area bounded by the curve $y=\sin x$ and $y=\cos x$ in between $x=0$ and $x=\frac{\pi }{2}$ is

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

Q. Let origin is one vertex of an equilateral triangle of side length $a$ units. If other vertex lies on the line $x-\sqrt{3}y=0$ in the first quadrant, then the co-ordinates of third vertex is/are

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

Q.

Sketch the region {(x, 0) : y = $\sqrt{4-x^2}$ and X - axis. Find the area of the region using integration.

Q.

For all values of $\theta$, the lines represented by the equation
$(2cos\theta +3sin\theta )x+(3cos\theta -5sin\theta )y-(5cos\theta -2sin\theta )=0$

1. pass through a fixed point

2. pass through the point (1, 1)

3. pass through a fixed point whose reflection in the line
   $x+y=\sqrt{2}$ is$(\sqrt{2}-1,\sqrt{2}-1)$

4. pass through the origin

Q.

Find the distance of the point (2, 5) from the line 3 x + y + 4 = 0 measured parallel to a line having slope $3/4.$

Q. The equation(s) of angular bisector between the intersecting lines $L_1:\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and $L_2:\frac{x}{-3}=\frac{y}{-2}=\frac{z}{-1}$ is/are

1. $\frac{y}{2}=\frac{z}{3},x=0$
2. $x+z=y=0$
3. $x=y=z$
4. $\frac{x}{1}=\frac{z}{2},y=0$

Q. The area of the part of the circle $x^2+y^2=8a^2$ and the parabola $y^2=2ax$ through which positive X-axis passes is

1. $4a^2\left(\frac{3\pi +2}{3}\right)$
2. $\left(\frac{a^2(3\pi +2)}{3}\right)$
   
3. $2a^2\left(\frac{3\pi -2}{3}\right)$
4. $\left(\frac{2a^2(3\pi +2)}{3}\right)$

Q.

Find the equations to the straight lines passing through the point (2, 3) and inclined at an angle of ${45}^{\circ }$ to the line $3x+y-5=0$

Q.

The straight line through $(P{x}_1,y_1)$ inclined at an angle $\theta$ with the x-axis meets the line ax + by + c = 0 in Q. Find the length of PQ.

Q. A point on the line $(x,y)=(-3+r\cos \frac{\pi }{6},2+r\sin \frac{\pi }{6})$ which is at a distance of $2$ units from the point $(-3,2)$ is

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