GATE 2020 IN Q.28: solution and answer

GATE 2020 · INQ.28 · 2 marks · MCQ

A straight line drawn on an x-y plane intercepts the x-axis at $- 0.5$ and the y-axis at $1$ . The equation that describes this line is ____

(A)
$y = - 0.5 x + 1$
(B)
$y = x - 0.5$
(C)
$y = 0.5 x - 1$
(D)
$y = 2 x + 1$

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Answer and solution

ANSWER

The line is $y = 2 x + 1$ , which is option (D).

Official answer: DSource: Official answer key, GATE 2020 IN (IIT Delhi)

Use the intercept form of a straight line, $\frac{x}{a} + \frac{y}{b} = 1$ , where $a$ is the x-intercept and $b$ is the y-intercept. Here $a = - 0.5$ and $b = 1$ .
Substitute: $\frac{x}{- 0.5} + \frac{y}{1} = 1 \Rightarrow - 2 x + y = 1 \Rightarrow y = 2 x + 1$
Cross-check with the slope-intercept form $y = m x + c$ . The line passes through $(- 0.5, 0)$ and $(0, 1)$ , so $m = \frac{1 - 0}{0 - ( - 0.5 )} = 2$ , and $c = 1$ (the y-intercept). Again $y = 2 x + 1$ .
Check the intercepts of $y = 2 x + 1$ : putting $x = 0$ gives $y = 1$ , and putting $y = 0$ gives $x = - 0.5$ . Both match the question.
Rule out the other options by their intercepts. (A) $y = - 0.5 x + 1$ cuts the x-axis at $x = 2$ , not $- 0.5$ . (B) $y = x - 0.5$ cuts the y-axis at $- 0.5$ , not $1$ . (C) $y = 0.5 x - 1$ cuts the y-axis at $- 1$ , not $1$ . The answer is (D).

KEY IDEA

A line with x-intercept $a$ and y-intercept $b$ (both non-zero) is $\frac{x}{a} + \frac{y}{b} = 1$ . Its slope is $- \frac{b}{a}$ and the y-intercept is $b$ itself. Here the slope is $- \frac{1}{- 0.5} = 2$ and $c = 1$ .

WATCH OUT FOR

Taking the slope as $b / a$ or $a / b$ without the minus sign, or using the x-intercept value $- 0.5$ as the slope. Both lead to option (A) $y = - 0.5 x + 1$ . That option has the right y-intercept, but it cuts the x-axis at $x = 2$ . Always check both intercepts of the option you pick.

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GATE 2020 IN Q.28: solution and answer

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