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UPSC NDA 2 2021 Mathematics Solved Paper

Question Numbers: 57-58

Direction: Consider the following for the next two (02) items that follow.

The equations of the sides AB, BC and CA of a triangle ABC are x - 2 = 0, y + 1 = 0 and x + 2y - 4 = 0 respectively.

Question : 57 of 120

Marks: +1, -0

What is the equation of the altitude through B on AC?

Solution:

Concept:

Altitude from B to AC is perpendicular to AC.

Product of slopes of perpendicular lines

= -1

We use point-slope form:

y - y_1 = m(x - x_1)

Explanation:

From the figure, slope of AC is

-\frac{1}{2}

Let slope of altitude BD be

m

Since

BD \perp AC

\left(-\frac{1}{2}\right) \times m = -1 \Rightarrow m = 2

Find coordinates of B:

Intersection of lines AB (

x - 2 = 0

) and BC (

y + 1 = 0

) gives

x = 2

y = -1

Thus B =

(2, -1)

Equation of altitude through B (slope

2

(y + 1) = 2(x - 2)

\Rightarrow y + 1 = 2x - 4

\Rightarrow 2x - y - 5 = 0

Answer:

2x - y - 5 = 0

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