Question

In the xy‑plane above, a point (not shown) with coordinates (s, t) lies on the graph of the linear function f. If s and t are positive integers, what is the ratio of t to s ?

A) 1 to 3
B) 1 to 2
C) 2 to 1
D) 3 to 1​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

• In the xy‑plane above, a point (not shown) with coordinates (s, t) lies on the graph of the linear function f and s and t are positive integers.

From graph, we concluded that the line passes through two points (0, 0) and (3, 6).

We know,

Equation of line passing through the points (a, b) and (c, d) is given by

$\rm :\longmapsto\:\boxed{\tt{ \: \: y - b = \dfrac{d - b}{c - a}(x - a) \: \: }}$

So, here

$\rm :\longmapsto\:a = 0$

$\rm :\longmapsto\:b = 0$

$\rm :\longmapsto\:c = 3$

$\rm :\longmapsto\:d = 6$

So, equation of line is

$\rm :\longmapsto\:y - 0 = \dfrac{6 - 0}{3 - 0} (x - 0)$

$\rm :\longmapsto\:y = \dfrac{6}{3} (x)$

$\rm :\longmapsto\:y = 2x$

Now, Coordinates (s, t) lies on this linear function.

So,

$\rm :\longmapsto\:t = 2s$

$\bf\implies \:\dfrac{t}{s} = \dfrac{2}{1}$

$\bf\implies \:t : s \: = \: 2 : 1$

So, Option (C) is correct.

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Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.