General Form of a Straight Line

Q. Let $A$ be the set of all points $(\alpha ,\beta )$ such that the area of triangle formed by the points $(5,6),(3,2)$ and $(\alpha ,\beta )$ is $12$ square units. Then the least possible length of a line segment joining the origin to a point in $A,$ is

1. $\frac{8}{\sqrt{5}}$
2. $\frac{16}{\sqrt{5}}$
3. $\frac{4}{\sqrt{5}}$
4. $\frac{12}{\sqrt{5}}$

Q. If the coordinates of the points A and B be (3, 3) and (7, 6), then the length of the portion of the line AB intercepted between the axes is

1. None of these
2. $\frac{5}{4}$
3. $\frac{\sqrt{10}}{4}$
4. $\frac{\sqrt{13}}{3}$

Q. For specifying a straight line how many geometrical parameters should be known

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

Q. Equation of the hour hand at 4 O’ clock is

1. $x-\sqrt{3}y=0$
2. $\sqrt{3}x-y=0$
3. $x+\sqrt{3}y=0$
4. $\sqrt{3}x+y=0$

Q. For $a>b>c>0$, the distance between (1, 1) and the point of intersection of the lines $ax+by+c=0$ and $bx+ay+c=0$ is less then $2\sqrt{2}$. Then

1. $a+b-c>0$
2. $a-b+c<0$
3. $a-b+c>0$
4. $a+b-c<0$

Q. Let $a,b,c$ be real numbers with $a^2+b^2+c^2=1$ Show that the equation $\left|\begin{array}{ccc}ax-by-c& bx+ay& cx+a\\ bx+ay& -ax+by-c& cy+b\\ cx+a& cy+b& -ax-by+c\end{array}\right|$ = 0
represents a straight line.

Q.

For $a>b>c>0$, the distance between (1, 1) and the point of intersection of the lines $ax+by+c=0$ and $bx+ay+c=0$ is less than $2\sqrt{2}$. Then,

1. $a+b-c>0$

2. $a-b+c<0$

3. $a-b+c>0$

4. $a+b-c<0$

Q. If the equation of the locus of a point equidistant from the points $(a_1,b_1)and(a_2,b_2)is(a_1-a_2)x+(b_1-b_2)y+c=0$
then the value of c is

1. $\sqrt{a_1^2+b_2^2-a_2^2-b_2^2}$
2. $\frac{1}{2}(a_2^2+b_2^2-a_1^2-b_1^2)$
3. $a_1^2+a_2^2+b_1^2-b_2^2$
4. $\frac{1}{2}(a_1^2+a_2^2-b_1^2-b_2^2)$

Q. If a, b, and c are in AP, then the straight line $ax+by+c=0$ will always pass through a fixed point whose coordinates are___

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

Q. The straight lines x + y = 0, 3x + y- 4 = 0 and x + 3y - 4 = 0 form a triangle which is

1. equilateral
2. isosceles
3. Right-angled
4. None of these