Straight Lines · JEE Main previous year questions

Let the area of the triangle with vertices $\mathrm{A}(1,\alpha )$, $\mathrm{B}(\alpha ,0)$ and $\mathrm{C}(0,\alpha )$ be 4 sq. units. If the points $(\alpha ,-\alpha )$, $(-\alpha ,\alpha )$ and $({\alpha }^2,\beta )$ are collinear, then $\beta$ is equal to :

1. A.64
2. B.$-8$
3. C.$-64$
4. D.512

The slope of normal at any point $(x,y)$, $x>0$, $y>0$ on the curve $y=y(x)$ is given by $\frac{x^2}{xy-x^2{y}^2-1}$. If the curve passes through the point $(1,1)$, then $e\cdot y(e)$ is equal to

1. A.$\frac{1-\tan (1)}{1+\tan (1)}$
2. B.$\tan (1)$
3. C.1
4. D.$\frac{1+\tan (1)}{1-\tan (1)}$

1. A.${\tan }^{-1}\left(\frac{3}{4}\right)$
2. B.${\tan }^{-1}(1)$
3. C.${\tan }^{-1}\left(\frac{4}{3}\right)$
4. D.${\tan }^{-1}\left(\frac{1}{2}\right)$

Let the equation of the pair of lines, y = px and y = qx, can be written as $(y-px)(y-qx)=0$. Then the equation of the pair of the angle bisectors of the lines $x^2-4xy-5y^2=0$ is :

1. A.$x^2-3xy+y^2=0$
2. B.$x^2+4xy-y^2=0$
3. C.$x^2+3xy-y^2=0$
4. D.$x^2-3xy-y^2=0$

Let a curve y = f(x) pass through the point $(2,({\log }_{e}2{)}^2)$ and have slope $\frac{2y}{x{\log }_{e}x}$ for all positive real value of x. Then the value of f(e) is equal to ____.

The equation of one of the straight lines which passes through the point $(1,3)$ and makes an angles ${\tan }^{-1}\left(\sqrt{2}\right)$ with the straight line, $y+1=3\sqrt{2}x$ is :

1. A.$4\sqrt{2}x+5y-\left(15+4\sqrt{2}\right)=0$
2. B.$5\sqrt{2}x+4y-\left(15+4\sqrt{2}\right)=0$
3. C.$4\sqrt{2}x+5y-4\sqrt{2}=0$
4. D.$4\sqrt{2}x-5y-\left(5+4\sqrt{2}\right)=0$

If $\alpha ,\beta$ are natural numbers such that $100^{\alpha }-199\beta =(100)(100)+(99)(101)+(98)(102)+\cdots +(1)(199)$, then the slope of the line passing through $(\alpha ,\beta )$ and origin is :

1. A.540
2. B.550
3. C.530
4. D.510

The intersection of three lines $x-y=0$, $x+2y=3$ and $2x+y=6$ is a :

1. A.Right angled triangle
2. B.Equilateral triangle
3. C.None of the above
4. D.Isosceles triangle

A ray of light coming from the point $(2,2\sqrt{3})$ is incident at an angle $30^{\circ }$ on the line $x=1$ at the point A. The ray gets reflected on the line $x=1$ and meets x-axis at the point B. Then, the line AB passes through the point :

1. A.$(3,-\sqrt{3})$
2. B.$(4,-\sqrt{3})$
3. C.$\left(4,-\frac{\sqrt{3}}{2}\right)$
4. D.$\left(3,-\frac{1}{\sqrt{3}}\right)$

If a $△\mathrm{ABC}$ has vertices $\mathrm{A}(-1,7)$, $\mathrm{B}(-7,1)$ and $\mathrm{C}(5,-5)$, then its orthocentre has coordinates :

1. A.$(-3,3)$
2. B.$(3,-3)$
3. C.$\left(\frac{3}{5},-\frac{3}{5}\right)$
4. D.$\left(-\frac{3}{5},\frac{3}{5}\right)$

Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y – 1 = 0 and 3x – y + 1 = 0. Then the line passing through the points C and P also passes through the point :

1. A.(–9, –7)
2. B.(9, 7)
3. C.(7, 6)
4. D.(–9, –6)

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

1. A.3x - 2y = 0
2. B.7x - 5y = 0
3. C.2x - 3y = 0
4. D.5x - 7y = 0

The equation $y=\sin x\sin (x+2)-{\sin }^2(x+1)$ represents a straight line lying in :

1. A.first, second and fourth quadrants
2. B.first, third and fourth quadrants
3. C.second and third quadrants only
4. D.third and fourth quadrants only

A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of $60^{\circ }$ with the line $x+y=0$. Then an equation of the line L is :

1. A.$x+\sqrt{3}y=8$
2. B.$\sqrt{3}x+y=8$
3. C.$(\sqrt{3}+1)x+(\sqrt{3}-1)y=8\sqrt{2}$
4. D.$(\sqrt{3}-1)x+(\sqrt{3}+1)y=8\sqrt{2}$

A triangle has a vertex at $(1,2)$ and the mid points of the two sides through it are $(-1,1)$ and $(2,3)$. Then the centroid of this triangle is :

1. A.$\left(\frac{1}{3},2\right)$
2. B.$\left(\frac{1}{3},\frac{5}{3}\right)$
3. C.$\left(1,\frac{7}{3}\right)$
4. D.$\left(\frac{1}{3},1\right)$

If the line ax + y = c, touches both the curves $x^2+y^2=1$ and $y^2=4\sqrt{2}x$ , then |c| is equal to :

1. A.2
2. B.$\sqrt{2}$
3. C.$\frac{1}{\sqrt{2}}$
4. D.$\frac{1}{2}$

Slope of a line passing through P(2, 3) and intersecting the line, x + y = 7 at a distance of 4 units from P, is :

1. A.$\frac{\sqrt{7}-1}{\sqrt{7}+1}$
2. B.$\frac{\sqrt{5}-1}{\sqrt{5}+1}$
3. C.$\frac{1-\sqrt{5}}{1+\sqrt{5}}$
4. D.$\frac{1-\sqrt{7}}{1+\sqrt{7}}$

If the straight line, $2x-3y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15,\beta )$, then $\beta$ equals :

1. A.$\frac{35}{3}$
2. B.$-5$
3. C.$-\frac{35}{3}$
4. D.5

If a straight line passing through the point P(–3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is :

1. A.x – y + 7 = 0
2. B.4x – 3y + 24 = 0
3. C.4x + 3y = 0
4. D.3x – 4y + 25 = 0

The locus of the point of intersection of the lines, $\sqrt{2}x-y+4\sqrt{2}k=0$ and $\sqrt{2}kx+ky-4\sqrt{2}=0$ (k is any non-zero real parameter), is :

1. A.an ellipse whose eccentricity is $\frac{1}{\sqrt{3}}$.
2. B.an ellipse with length of its major axis $8\sqrt{2}$.
3. C.a hyperbola whose eccentricity is $\sqrt{3}$.
4. D.a hyperbola with length of its transverse axis $8\sqrt{2}$.

If the angle between the lines, $\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\frac{5-x}{-2}=\frac{7y-14}{p}=\frac{z-3}{4}$ is ${\cos }^{-1}\left(\frac{2}{3}\right)$, then p is equal to :

1. A.$\frac{7}{2}$
2. B.$\frac{2}{7}$
3. C.$-\frac{7}{4}$
4. D.$-\frac{4}{7}$

A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is :

1. A.3x + 2y = 6xy
2. B.3x + 2y = 6
3. C.2x + 3y = xy
4. D.3x + 2y = xy

The locus of the point of intersection of the straight lines, $tx-2y-3t=0$ $x-2ty+3=0$ ($t\in \mathbf{R}$), is :

1. A.an ellipse with eccentricity $\frac{2}{\sqrt{5}}$
2. B.an ellipse with the length of major axis $6$
3. C.a hyperbola with eccentricity $\sqrt{5}$
4. D.a hyperbola with the length of conjugate axis $3$

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0,0)$, $(0,41)$ and $(41,0)$ is :

1. A.$820$
2. B.$780$
3. C.$901$
4. D.$861$

Locus of the image of the point $(2,3)$ in the line $(2x-3y+4)+k(x-2y+3)=0$, $k\in R$, is a :

1. A.circle of radius $\sqrt{2}$.
2. B.circle of radius $\sqrt{3}$.
3. C.straight line parallel to $x$-axis
4. D.straight line parallel to $y$-axis