Concurrence of three lines; distance of a point from a line — practice questions

Question 1 · difficulty L2 · understanding

The image of the point (3, 5) in the line x $$-$$ y + 1 = 0, lies on :

• A. (x $$-$$ 4) 2 + (y $$-$$ 4) 2 = 8
• B. (x $$-$$ 4) 2 + (y $$+$$ 2) 2 = 16
• C. (x $$-$$ 2) 2 + (y $$-$$ 2) 2 = 12
• D. (x $$-$$ 2) 2 + (y $$-$$ 4) 2 = 4

Question 2 · difficulty L3 · understanding

If the sum of squares of all real values of $$\alpha$$, for which the lines $$2 x-y+3=0,6 x+3 y+1=0$$ and $$\alpha x+2 y-2=0$$ do not form a triangle is $$p$$, then the greatest integer less than or equal to $$p$$ is _________.

Question 3 · difficulty L3 · understanding

Let the equations of two adjacent sides of a parallelogram $$\mathrm{ABCD}$$ be $$2 x-3 y=-23$$ and $$5 x+4 y=23$$. If the equation of its one diagonal $$\mathrm{AC}$$ is $$3 x+7 y=23$$ and the distance of A from the other diagonal is $$\mathrm{d}$$, then $$50 \mathrm{~d}^{2}$$ is equal to ____________.

Question 4 · difficulty L3 · understanding

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $x+2 \sqrt{2} y=4$. If the co-ordinates of the vertex A are $(\alpha, \beta)$, then the greatest integer less than or equal to $|\alpha+\sqrt{2} \beta|$ is

• A. 5
• B. 4
• C. 2
• D. 3

Question 5 · difficulty L3 · understanding

A rectangle is formed by the lines $x=0, y=0, x=3$ and $y=4$. Let the line L be perpendicular to $3 x+y+6=0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $\left(\frac{1}{2},-5\right)$ from the line $L$ is equal to :

• A. $\sqrt{10}$
• B. $2 \sqrt{5}$
• C. $2 \sqrt{10}$
• D. $3 \sqrt{10}$

Question 6 · difficulty L3 · understanding

Let a point A lie between the parallel lines $\mathrm{L}_1$ and $\mathrm{L}_2$ such that its distances from $\mathrm{L}_1$ and $\mathrm{L}_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C lie on the lines $\mathrm{L}_1$ and $\mathrm{L}_2$, respectively, is :

• A. $21 \sqrt{3}$
• B. $12 \sqrt{2}$
• C. $15 \sqrt{6}$
• D. 27

Question 7 · difficulty L3 · understanding

Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $$\alpha$$ with the positive x-axis and the equations of its diagonals are $(\sqrt{3}+1)x+(\sqrt{3}-1)y=0$ and $(\sqrt{3}-1)x-(\sqrt{3}+1)y+8\sqrt{3}=0$. Then $a$ 2 is equal to :

• A. 48
• B. 16
• C. 24
• D. 32

Question 8 · difficulty L3 · understanding

Consider the lines $x(3 \lambda+1)+y(7 \lambda+2)=17 \lambda+5, \lambda$ being a parameter, all passing through a point P. One of these lines (say $L$ ) is farthest from the origin. If the distance of $L$ from the point $(3,6)$ is $d$, then the value of $d^2$ is

• A. 10
• B. 20
• C. 15
• D. 30

Question 9 · difficulty L3 · understanding

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $\mathrm{L}_1: 2 x+y+6=0$ and $\mathrm{L}_2: 4 x+2 y-p=0, p>0$, at the points A and B , respectively. If $A B=\frac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point $A$ on the line $L_2$ is $M$, then $\frac{A M}{B M}$ is equal to

• A. 5
• B. 3
• C. 2
• D. 4

Question 10 · difficulty L3 · understanding

The vertices of a triangle are $$\mathrm{A}(-1,3), \mathrm{B}(-2,2)$$ and $$\mathrm{C}(3,-1)$$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

• A. $$-x+y-(2-\sqrt{2})=0$$
• B. $$x+y-(2-\sqrt{2})=0$$
• C. $$x+y+(2-\sqrt{2})=0$$
• D. $$x-y-(2+\sqrt{2})=0$$