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Official PYQ · 2020QGA-9MCQ2 marks

GATE CS 2020 QGA-9 — General Aptitude

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Two straight lines are drawn perpendicular to each other in X−YX-Y plane. If α\alpha and β\beta are the acute angles the straight lines make with the X-\text{X-} axis, then α+β\alpha + \beta is ________.

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Answer: B

Let the inclinations of the two perpendicular lines with the positive X-axis be θ1\theta_1 and θ2\theta_2. Since the lines are perpendicular, their inclinations differ by 90∘90^\circ:

θ2=θ1+90∘\theta_2 = \theta_1 + 90^\circ

Assuming the lines are not parallel to the coordinate axes, let 0∘<θ1<90∘0^\circ < \theta_1 < 90^\circ. The acute angle α\alpha that the first line makes with the X-axis is:

α=θ1\alpha = \theta_1

The second line has an inclination θ2=θ1+90∘\theta_2 = \theta_1 + 90^\circ, which lies in the second quadrant (90∘<θ2<180∘90^\circ < \theta_2 < 180^\circ). The acute angle β\beta it makes with the X-axis is:

β=180∘−θ2=180∘−(θ1+90∘)=90∘−θ1\beta = 180^\circ - \theta_2 = 180^\circ - (\theta_1 + 90^\circ) = 90^\circ - \theta_1

Adding the two acute angles gives:

α+β=θ1+(90∘−θ1)=90∘\alpha + \beta = \theta_1 + (90^\circ - \theta_1) = 90^\circ

Thus, the sum of the acute angles is always 90∘90^\circ.

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