Family of Straight Lines

Q. A general equation of second degree i.e., $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ always represents a pair of straight lines.

1. True
2. False

Q. Let the points of intersections of the lines $x-y+1=0,x-2y+3=0$ and $2x-5y+11=0$ are the mid points of the sides of a triangle $ABC$. Then the area of the triangle $ABC$ is $\text{sq. units.}$

Q.

A triangle with vertices (4, 0), (-1, -1), (3, 5) is

1. Isosceles but not right angled

2. Right angled but not isosceles

3. Neither right angled nor isosceles

4. Isosceles and right angled

Q.

Find the equation of straight line which passes through the point (2, -3) and the point of intersection of the lines x+y +4 = 0 and 3x-y-8=0

1. x-2y-7 = 0

2. 2x-y-7 = 0

3. x-2y+7 = 0

4. 2x-y+7 = 0

Q.

The corner points of the feasible region determined by the following system of linear inequalities :

$2x+y\ge 10,x+3y\ge 15,x,y\ge 0are(0,0),(5,0),(3,4)and(0,5).$ Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is

(a) p = q

(b) p = 2q

(c) p = 3q

(d) q = 3p

Q.

How do you find slope with two variables?

Q. If the straight lines ax + by + p = 0 and $xcos\alpha +ysin\alpha -p=0$ include an angle $\frac{\pi }{4}$ between them and meet the
straight line $xsin\alpha -ycos\alpha =0$ in the same point, then the value of $a^2+b^2$ is equal to

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

Q. Given the family of lines $a(3x+4y+6)+b(x+y+2)=0.$ The line of the family situated at the greatest distance from the point $P(2,3),$ has the equation

1. $4x+3y+8=0$
2. $5x+3y+10=0$
3. $15x+8y+30=0$
4. $4x-3y+8=0$

Q. A family of lines is given by $(1+2\lambda )x+(1-\lambda )y+\lambda =0,$ $\lambda$ being the parameter. The line belonging to this family at the maximum distance from the point $(1,4)$ is

1. $33x+12y-7=0$
2. $33x+12y+7=0$
3. $12x+33y+7=0$
4. $12x+33y-7=0$

Q. If the lines $x+y=|a|$ and $ax-y=1$ intersect each other in the first quadrant, then the range of $a$ is

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

Q.

The equation of the line which cuts off the intercepts $2\mathrm{a}\sec \left(\mathrm{\theta }\right)$ and $2\mathrm{a}cosec\left(\mathrm{\theta }\right)$ on the axes is

1. $\mathrm{x}\sin \left(\mathrm{\theta }\right)+\mathrm{y}\cos \left(\mathrm{\theta }\right)–2\mathrm{a}=0$

2. $\mathrm{x}\cos \left(\mathrm{\theta }\right)+\mathrm{y}\sin \left(\mathrm{\theta }\right)–2\mathrm{a}=0$

3. $\mathrm{x}\sec \left(\mathrm{\theta }\right)+\mathrm{y}cosec\left(\mathrm{\theta }\right)–2\mathrm{a}=0$

4. $\mathrm{x}\csc \left(\mathrm{\theta }\right)+\mathrm{y}\sec \left(\mathrm{\theta }\right)-2\mathrm{a}=0$

Q.

A line $AB$ makes zero intercepts on x-axis and y-axis and it is perpendicular to another line $CD,3x+4y+6=0.$The equation of line $AB$ is

1. $y=4$

2. $4x–3y+8=0$

3. $4x–3y=0$

4. $4x–3y+6=0$

Q.

The equation of the normal to the curve $y={(1+x)}^{2y}+{\cos }^2\left({\sin }^{-1}\left(x\right)\right)$ at $x=0$ is

1. $y+4x=2$

2. $2y+x=4$

3. $x+4y=8$

4. $y=4x+2$

Q. Consider a family of straight lines $(x+y)+\lambda (2x-y+1)=0$. Then the equation of the straight line belonging to this family that is farthest from $(1,-3)$ is:

1. $6x+15y+7=0$
2. $6x-15y+7=0$
3. $6x+15y+5=0$
4. $6x-15y+5=0$

Q. A diagonal of rhombus $ABCD$ is member of both the families of lines
$(x+y-1)+{\lambda }_1(2x+3y-2)=0$ and
$(x-y+2)+{\lambda }_2(2x-3y+5)=0$
where ${\lambda }_1$ and ${\lambda }_2\in R$ and one of the vertex of rhombus is $(3,2)$. If area of the rhombus is $12\sqrt{5}$ square units, then the length of the longer diagonal of the rhombus is

Q. If the coordinates of the four vertices of a quadrilateral are $(-2,4),(-1,2),(1,2)$ and $(2,4)$ taken in order, then the equation of line passing through the vertex $(-1,2)$ and dividing the quadrilateral in two equal areas is

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

Q. Let a and b be non-zero and real numbers. Then, the equation
$(a{x}^2+b{y}^2+c)(x^2-5xy+6y^2)=0$ represents

1. Four straight lines, when c=0 and a, b are of the same sign
2. Two straight lines and a circle, when a=b and c is of sign opposite to that of a
3. Two straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a
4. A circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

Q. The set of values of $\alpha$ for which the point $P(\alpha ,{\alpha }^2-2)$ lies inside the triangle formed by the lines $x+y=1,$ $y=x+1$ and $y=-1,$ is

1. $(\frac{1-\sqrt{13}}{2},-1)\cup (1,\frac{-1+\sqrt{13}}{2})$
2. $(-\sqrt{3},\sqrt{3})$
3. $(\frac{1-\sqrt{13}}{2},\frac{-1+\sqrt{13}}{2})$
4. $[-1,1]$

Q. Draw a quadrilateral in the Cartesian plane, whose vertices are $(–4,5),(0,7),(5,–5)\text{and}(–4,–2)$. Also, find its area.

Q. A general equation of second degree i.e., $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ always represents a pair of straight lines.

1. True
2. False

Q. The equation of the straight line passing through the point of intersection of the lines $x-y=1$ and $2x-3y+1=0$ and parallel to the line $3x+4y=14$ is

1. $3x+4y+24=0$
2. $4x+3y-24=0$
3. $3x+4y-24=0$
4. $4x+3y+24=0$

Q. If $a^2-b^2-c^2-2bc=0,$ then the family of lines $ax+by+c=0$ are concrrent at the points

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

Q. The equation of the straight line through the intersection of line $2x+y=1$ and $3x+2y=5$ and passes through the origin is

1. $7x+3y=0$
2. $7x-y=0$
3. $3x+2y=0$
4. $3x+y=0$

Q.

If $a+b+c=0,$ then the family of lines $3ax+by+2c=0$ pass through fixed point

1. $(2,\frac{2}{3})$

2. $(\frac{2}{3},2)$

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

4. none of these

Q. Let line $L$ passes through the point of intersection of $2x+y-1=0$ and $x+2y-2=0$. If $L$ makes a triangle with the coordinate axes of area $\frac{5}{8}\text{sq. units}$, then the equation of $L$ can be

1. $4x-5y+5=0$
2. $4x+5y-5=0$
3. $4x-5y-5=0$
4. $4x+5y+5=0$

Q. Let $\overrightarrow{AB}=3\hat{i}–\hat{j}+\hat{k}$ and $\overrightarrow{CD}=3\hat{i}+2\hat{j}+4\hat{k}$ are two vectors. The position vectors of the points $\overrightarrow{A}$ and $\overrightarrow{C}$ are $6\hat{i}+7\hat{j}+4\hat{k}$ and $–9\hat{j}+2\hat{k}$, respectively. Find the position vector of a point $P$ on the line $AB$ and a point $Q$ on the line $CD$ such that $\overrightarrow{PQ}$ is perpendicular to the vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$.

Q.

Prove that the family of lines represented by $x(1+\lambda )+y(2-\lambda )+5=0,\lambda$ being arbitary, pass through a fixed point .Also, find the fixed point.

Q. The family of lines $x(a+b)+y(a-b)=2a,a,b\in R$ are concurrent at $(p,q)$ then the value of $p+q$ is

Q. The $x$-intercept of angle bisector of angle between the lines $2x+y-2=0$ and $2x+4y+7=0$ which contains the fixed point on the family of lines $(2\cos \alpha +3\sin \alpha )x+(3\cos \alpha -5\sin \alpha )y=5\cos \alpha -2\sin \alpha$ for different values of $\alpha$, is equal to

1. $\frac{11}{2}$
2. $-\frac{11}{2}$
3. $-\frac{1}{2}$
4. $\frac{1}{2}$

Q. The equation of the straight line through the intersection of line $2x+y=1$ and $3x+2y=5$ and passes through the origin is

1. $7x+3y=0$
2. $7x-y=0$
3. $3x+2y=0$
4. $3x+y=0$