Pair of Lines Passing Though Origin

Q.

$a{x}^2+2hxy+b{y}^2=0$ always represents a pair of straight lines passing through the origin. If
$\begin{array}{cc}\text{Column 1}& \text{Column 2}\\ \text{a.}{h}^2>ab& \text{1. Lines are coincident}\\ \text{b.}{h}^2=ab& \text{2. Lines are real and distinct}\\ \text{c.}{h}^2<ab\text{3. Lines are imaginary with real point of intersection i.e. (0, 0)}& \end{array}$

1. a-1, b-2, c-3

2. a-2, b-1, c-3

3. a-3, b-2, c-1

4. a-3, b-1, c-2

Q. If one of the lines given by the equation $2x^2+pxy+3y^2=0,$ coincides with one of those given by $2x^2+qxy-3y^2=0$ and the other lines represented by them are perpendicular, then $(p,q)$ can be

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

Q. If $a{x}^3+b{y}^3+c{x}^{2}y+dx{y}^2=0$ represents three distinct straight lines such that each line bisects the angle between the other two, then which of the following is (are) CORRECT?

1. $3b+c=0$
2. $3c+b=0$
3. $d^2>3bc$
4. $d^2<3bc$

Q.

The straight lines represented by the equation $a{x}^2+2bxy+b{y}^2=0$ are perpendicular if

1. a+b = 0

2. a-b = 0

3. a+2h+b = 0

4. a = 2h

Q.

Find the equation of the lines joining the origin to the points of intersection of the straight line y=3x+2 with the curve $x^2+2xy+3y^2+4x+8y-11=0$

1. $7x^2-2xy-y^2=0$

2. $7x^2-2xy+y^2=0$

3. $7x^2+2xy+y^2=0$

4. $7x^2+2xy+y^2=0$

Q. If the pairs of lines $x^2+2xy+a{y}^2=0$ and $a{x}^2+2xy+y^2=0$ have exactly one line in common, then the combined equation of the other two lines is given by

1. $3x^2+8xy-3y^2=0$
2. $3x^2+10xy+3y^2=0$
3. $3x^2-2xy-y^2=0$
4. $x^2+2xy-3y^2=0$

Q.

Equation $6x^2-5xy+y^2=0$ represents____________________

1. Pair of coincident lines

2. Pair of imaginary lines with real point of intersection i.e.(0, 0)

3. None of these

4. Pair of distinct straight lines

Q. Let $L_1:2x+3y=5$ and $L_2$ be equal sides of a triangle. If $L:x+y=2$ is the perpendicular bisector of the third side, then the equation of $L_2$ is

1. $3x+y=4$
2. $x+5y=6$
3. $x=y$
4. $3x+2y=5$

Q. If the slope of one of the lines given by $a{x}^2+2hxy+b{y}^2=0$, be the square of the slope of the other, then $\frac{a+b}{h}+\frac{8h^2}{ab}$ is

Q.

If the lines represented by the equation $2x^2-3xy+y^2=0$ make angles $\alpha$ and $\beta$ with x - axis, then $co{t}^2\alpha +co{t}^2\beta =$

1. 0

2. 3/2

3. 7/4

4. 5/4

Q. What is the maximum slope of the curve $y=-x^3+3x^2+2x-27?$

1. $2$
2. $5$
3. $21$
4. $12$

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 $Q$, the length $PQ$ is

1. $\frac{132}{12\sqrt{3}-5}$
2. $\frac{132}{5\sqrt{3}-12}$
3. $\frac{132}{5\sqrt{3}+12}$
4. $\frac{132}{12\sqrt{3}+5}$

Q. The angle between the lines $\sqrt{3x}+y+1=0$ and $x+1=0$ is.

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

Q. If the pairs of lines $x^2+2xy+a{y}^2=0$ and $a{x}^2+2xy+y^2=0$ have exactly one line in common, then the combined equation of the other two lines is given by

1. $3x^2+8xy-3y^2=0$
2. $3x^2+10xy+3y^2=0$
3. $3x^2-2xy-y^2=0$
4. $x^2+2xy-3y^2=0$

Q. If one of the lines given by the equation $2x^2+pxy+3y^2=0,$ coincides with one of those given by $2x^2+qxy-3y^2=0$ and the other lines represented by them are perpendicular, then $(p,q)$ can be

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

Q. If $a{x}^3+b{y}^3+c{x}^{2}y+dx{y}^2=0$ represents three distinct straight lines such that each line bisects the angle between the other two, then which of the following is (are) CORRECT?

1. $3b+c=0$
2. $3c+b=0$
3. $d^2>3bc$
4. $d^2<3bc$

Q.

If the pairs of lines $x^2+2xy+a{y}^2=0$ and $a{x}^2+2xy+y^2=0$ have exactly one line in common then the joint equation of the other two lines is given by

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

2. $3x^2+10xy+3y^2=0$

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

4. $x^2+2xy-3y^2=0$

Q. If $A_1$ and $A_2$ respectively represents the area bounded by the curves $f(x,y)$ $4x^2\le y\le 3x$ and $g$ (x, y)$4x^2\le y\le |3x|$ then $A_1$ : $A_2$ equals:

1. 2:1
2. 1:2
3. 1:3
4. 3:1

Q. A unit vector in the $xy$-plane that makes an angle of ${45}^0$ with the vector $\hat{i}+\hat{j}$ and an angle of ${60}^0$ with the vector $3\hat{i}+4\hat{j}$ is

1. $\hat{i}$
2. $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
3. None of these
4. $\frac{\hat{i}-\hat{j}}{\sqrt{2}}$

Q. A tangent having slope of $-\frac{4}{3}$ to the ellipse $\frac{x^2}{18}+\frac{y^2}{32}=1$ intersects the major and minor axes in points $A$ and $B$ respectively. If $O$ is the origin, find the area of $△OAB$.

1. $12$
2. $24$
3. $48$
4. $60$

Q.

Equation $6x^2-5xy+y^2=0$ represents____________________

1. Pair of coincident lines

2. Pair of imaginary lines with real point of intersection i.e.(0, 0)

3. Pair of distinct straight lines

4. None of these

Q. $A$ straight line through the origin $O$ meets the parallel lines $4x+2y=9$ and $2x+y+6=0$ at points $P$ and $Q$ respectively, then point $O$ divides the segment $PQ$ in the ratio:

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

Q. Let $f:R\to R$ be a positive increasing function with $\underset{x\to \infty }{lim}\frac{f\left(3x\right)}{f\left(x\right)}=1$. Then $\underset{x\to \infty }{lim}\frac{f\left(2x\right)}{f\left(x\right)}$

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

Q.

If the pairs of lines $x^2+2xy+a{y}^2=0$ and $a{x}^2+2xy+y^2=0$ have exactly one line in common then the joint equation of the other two lines is given by

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

2. $3x^2+10xy+3y^2=0$

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

4. $x^2+2xy-3y^2=0$

Q.

If the lines represented by the equation $2x^2-3xy+y^2=0$ make angles $\alpha$ and $\beta$ with x - axis, then $co{t}^2\alpha +co{t}^2\beta =$

1. 0

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

3. $\frac{7}{4}$

4. $\frac{5}{4}$

Q. In the symmetrical representation of three or more terms helps to solve sum or product of three terms so that in one of the equations ______

1. a vanishes.
2. The equation is no longer valid.
3. d completely vanishes.
4. a and d both vanish.

Q. Find the measure of acute angle between the lines represented by
$2x^2+7xy+3y^2=0$

Q. Find the angle of intersection of the curves:

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

Q. A circle is described with minor axis of the ellipse as diameter. If the foci lie on the circle, then the eccentricty of the ellipse is

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

Q. The slope of the line $|z1|=|z+i|$ is

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