Cartesian coordinates; distance formula; section formula — practice questions

Question 1 · difficulty L3 · understanding

Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $P Q R$ is formed such that $Q$ lies on one of the parallel lines, while R lies on the other. Then $(Q R)^2$ is equal to _________.

Question 2 · difficulty L3 · understanding

A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be ($$\alpha$$, $$\beta$$). Then, the value of 7$$\alpha$$ + 3$$\beta$$ is equal to ____________.

Question 3 · difficulty L3 · understanding

A man starts walking from the point P($$-$$3, 4), touches the x-axis at R, and then turns to reach at the point Q(0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $$50\left( {{{(PR)}^2} + {{(RQ)}^2}} \right)$$ is equal to ____________.

Question 4 · difficulty L3 · understanding

The equations of two sides $$\mathrm{AB}$$ and $$\mathrm{AC}$$ of a triangle $$\mathrm{ABC}$$ are $$4 x+y=14$$ and $$3 x-2 y=5$$, respectively. The point $$\left(2,-\frac{4}{3}\right)$$ divides the third side $$\mathrm{BC}$$ internally in the ratio $$2: 1$$, the equation of the side $$\mathrm{BC}$$ is

• A. $$x+6 y+6=0$$
• B. $$x-3 y-6=0$$
• C. $$x+3 y+2=0$$
• D. $$x-6 y-10=0$$

Question 5 · difficulty L3 · understanding

The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle $${\pi \over 4}$$ at the origin, is equal to :

• A. 10
• B. $${48 \over 5}$$
• C. $${52 \over 5}$$
• D. 3

Question 6 · difficulty L3 · understanding

Let the area of the triangle formed by a straight line $\mathrm{L}: x+\mathrm{b} y+\mathrm{c}=0$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of $45^{\circ}$ with the positive $x$-axis, then the value of $\mathrm{b}^2+\mathrm{c}^2$ is :

• A. 90
• B. 83
• C. 93
• D. 97

Question 7 · difficulty L3 · understanding

A variable line $$\mathrm{L}$$ passes through the point $$(3,5)$$ and intersects the positive coordinate axes at the points $$\mathrm{A}$$ and $$\mathrm{B}$$. The minimum area of the triangle $$\mathrm{OAB}$$, where $$\mathrm{O}$$ is the origin, is :

• A. 35
• B. 25
• C. 30
• D. 40

Question 8 · difficulty L3 · understanding

Let $$B$$ and $$C$$ be the two points on the line $$y+x=0$$ such that $$B$$ and $$C$$ are symmetric with respect to the origin. Suppose $$A$$ is a point on $$y-2 x=2$$ such that $$\triangle A B C$$ is an equilateral triangle. Then, the area of the $$\triangle A B C$$ is :

• A. $$\frac{10}{\sqrt{3}}$$
• B. $$2 \sqrt{3}$$
• C. $$3 \sqrt{3}$$
• D. $$\frac{8}{\sqrt{3}}$$

Question 9 · difficulty L3 · understanding

Let $$A(1,1), B(-4,3), C(-2,-5)$$ be vertices of a triangle $$A B C, P$$ be a point on side $$B C$$, and $$\Delta_{1}$$ and $$\Delta_{2}$$ be the areas of triangles $$A P B$$ and $$A B C$$, respectively. If $$\Delta_{1}: \Delta_{2}=4: 7$$, then the area enclosed by the lines $$A P, A C$$ and the $$x$$-axis is :

• A. $$\frac{1}{4}$$
• B. $$\frac{3}{4}$$
• C. $$\frac{1}{2}$$
• D. 1

Question 10 · difficulty L3 · understanding

Let $$\alpha$$ 1 , $$\alpha$$ 2 ($$\alpha$$ 1 < $$\alpha$$ 2 ) be the values of $$\alpha$$ fo the points ($$\alpha$$, $$-$$3), (2, 0) and (1, $$\alpha$$) to be collinear. Then the equation of the line, passing through ($$\alpha$$ 1 , $$\alpha$$ 2 ) and making an angle of $${\pi \over 3}$$ with the positive direction of the x-axis, is :

• A. $$x - \sqrt 3 y - 3\sqrt 3 + 1 = 0$$
• B. $$\sqrt 3 x - y + \sqrt 3 + 3 = 0$$
• C. $$x - \sqrt 3 y + 3\sqrt 3 + 1 = 0$$
• D. $$\sqrt 3 x - y + \sqrt 3 - 3 = 0$$