Pair of Lines

Q.

The equation $4x^2+12xy+9y^2+2gx+2fy+c=0$ will represent two real parallel straight lines if

1. $g=4,f=9,c=0$

2. $g=2,f=3,c=1$

3. $g=2,f=3,c$ is any number

4. $g=4,f=9,c>1$

Q.

Pair of straight lines perpendicular to each other represented by

1. $2x^2=2y(2x+y)$

2. $x^2+y^2+3=0$

3. $2x^2=y(2x+y)$

4. $x^2=2(x–y)$

Q. Let $ABC$ be a triangle with $A(-3,1)$ and $\angle ACB=\theta ,0<\theta <\frac{\pi }{2}.$ If the equation of the median through $B$ is $2x+y-3=0$ and the equation of angle bisector of $C$ is $7x-4y-1=0,$ then $\tan \theta$ is equal to

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

Q. If $2x^2+7xy+3y^2+8x+14y+\lambda =0$ represents a pair of straight lines, then the value of $\lambda$ is

Q. Let $a,b,c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies in the fourth quadrant and is equidistant from the two axes then:

1. $2bc-3ad=0$
2. $2bc+3ad=0$
3. $3bc-2ad=0$
4. $3bc+2ad=0$

Q.

The angle between the lines represented by the equation $\lambda x^2+{\left(1-\lambda \right)}^{2}xy-\lambda y^2=0$ is

1. $30°$

2. $45°$

3. $60°$

4. $90°$

Q.

The curve y = ax³ + bx² + cx + 5 touches the x-axis at P(-2, 0) and cuts the y-axis at the point Q where its gradient is 3. the equation of the curve is... .

1. 

2. 

3. 

4. 

Q. Find the equations of the lines parallel to axes and passing through $(–2,3)$.

Q. The angle between the lines 3x + 2y + z = 0 = x + y – 2z and 2x – y – z = 0 = 7x + 10y – 8z is :

1. $\frac{\pi }{6}$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{3}$
4. $0$

Q. The equations $x-y=4$ and $x^2+4xy+y^2=0$ represent the sides of

1. an equilateral triangle
2. a right angled triangle
3. an isosceles triangle
4. a right angled isosceles triangle

Q. If the adjacent sides of a parallelogram are represented by $2x^2-5xy+3y^2=0$ and the equation of one diagonal is $x+y-2=0,$ then the equation of the other diagonal is

1. $11x+9y=0$
2. $9x+11y=0$
3. $9x-11y=0$
4. $11x-9y=0$

Q. The distance between the pair of parallel lines represented by $x^2+4xy+4y^2+3x+6y-4=0$ is ___ units

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

Q. The combined equation of two sides of a triangle is $x^2-3y^2-2xy+8y-4=0.$ The third side, which is variable always passes through the point $(-5,-1)$. If the range of values of the slope of the third line such that the origin is an interior point of the triangle is $(a,b)$, then the value of $(a+\frac{1}{b})$ is

1. $\frac{1}{2}$
2. $2$
3. $4$
4. $0$

Q.

The lines $(lx+my{)}^2-3(mx-ly{)}^2=0$ and lx + my + n = 0 form

1. A right angled triangle

2. None of these

3. An isosceles triangle

4. An equilateral triangle

Q. Product of perpendicular distances drawn from origin to pair of straight lines $12x^2+25xy+12y^2+10x+11y+2=0$ is __ sq. units

1. $\frac{1}{25}$
2. $\frac{4}{25}$
3. $\frac{3}{25}$
4. $\frac{2}{25}$

Q.

The distance between the pair of parallel lines $x^2+2xy+y^2-8ax-8ay-9a^2=0$ is

1. $2\sqrt{5}a$

2. $\sqrt{10}a$

3. $10a$

4. $5\sqrt{2}$

Q. The area enclosed by the quadrilateral formed by $x^2{y}^2-9x^2-25y^2+225=0$ is

1. $60$ sq.units
2. $100$ sq.units
3. $30$ sq.units
4. $20$ sq.units

Q. The condition to be imposed on $\beta$ so that $(0,\beta )$ lies on or inside the triangle having sides y + 3x + 2 = 0, 3y – 2x – 5 = 0 and 4y + x – 14 = 0 is

1. 
2. 
3. 
4. 

Q. Area bounded by the ellipse $2x^2+3y^2=1$ is

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

Q. The straight lines represented by $(y-mx{)}^2=a^2(1+m^2)$ and $(y-nx{)}^2=a^2(1+n^2)$ forms
(where $mn\ne -1$ and $m\ne n$)

1. square
2. rhombus
3. rectangle
4. trapezium

Q.

Find the direction in which a straight line must be drawn through the point (–1, 2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.

Q.

The differential equation of the family of curves $\mathrm{y}=\mathrm{\alpha cos}\left(\mathrm{x}+\mathrm{b}\right)$ is

1. $\left(\frac{d^{2}y}{d{x}^2}\right)-y=0$

2. $\left(\frac{d^{2}y}{d{x}^2}\right)+y=0$

3. $\left(\frac{d^{2}y}{d{x}^2}\right)+2y=0$

4. None of these

Q.

Find the points on the x-axis, whose distances from the line are 4 units.

Q. If $y=mx$ bisects the angle between the lines $y=x^2({\sin }^2\theta +{\tan }^2\theta )+2xy\cos \theta +y^2{\sec }^2\theta =0$ when $\theta =\frac{\pi }{3}.$ If the value of $m$ is $\frac{a\pm \sqrt{b}}{4},$ then $b-10a=?$

Q. If area of quadrilateral formed by lines $2x^2+3xy-2y^2=0$ and $2x^2+3xy-2y^2+3x+y+1=0$ is $A$ sq. units, then the value of $20A$ is

Q. If the straight lines joining the origin and the point of the intersection of the curve $x^2+12xy-y^2+4x-2y+3=0$ and $x+ky-1=0$ are equally inclined to $x$-axis then the value of $k$ is

1. $1$
2. $-1$
3. $0$
4. $-2$

Q. Equation of a line which is parallel to the line common to the pair of lines given by $6x^2-xy-12y^2=0$ and $15x^2+14xy-8y^2=0$ and at a distance of $7$ units from it, is

1. $3x+4y=35$
2. $3x+4y+35=0$
3. $2x-3y=35$
4. $2x-3y+35=0$

Q.

Find the equation of the line which passes through the point (–4, 3) with slope.

Q. Find the equation of the circle drawn on the intercept made by the line 3x+4y=12 between the coordinate axes as diameter

Q. The lines joining the origin to the point of intersection of $3x^2+mxy-4x+1=0$ and $2x+y-1=0$ are at right angles. Then all possible values of $m$ lie in the interval

1. $(1,2]$
2. $[1,2]$
3. $\varphi$
4. $R$