Straight Line Examples

Example 1 : Find the equation of lines which passes through the point (3,4) and the sum of intercepts on the axes is 14.

Solution : Let the equation of line be $x\over a$ + $y\over b$ = 1   …..(i)

This line passes through (3,4), therefore $3\over a$ + $4\over b$ = 1   …….(ii)

It is given that a + b = 14   $\implies$   b = 14 – a in (ii), we get

$3\over a$ + $4\over 14 – a$ = 1   $\implies$   $a^2$ – 13a + 42 = 0

$\implies$   (a – 7)(a – 6) = 0   $\implies$   a = 7, 6

for a = 7, b = 14 – 7 = 7 and for a = 6, b = 14 – 6 = 8

Putting the values of a and b in (i), we get the equations of lines

$x\over 7$ + $y\over 7$ = 1   and   $x\over 6$ + $y\over 8$ = 1

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Example 2 : If x + 4y – 5 = 0 and 4x + ky + 7 = 0 are two perpendicular lines then k is –

Solution : $m_1$ = -$1\over 4$   $m_2$ = -$4\over k$

Two lines are perpendicular if $m_1 m_2$ = -1

$\implies$   (-$1\over 4$)$\times$(-$4\over k$) = -1   $\implies$   k = -1

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Example 3 : If the straight line 3x + 4y + 5 – k(x + y + 3) = 0 is parallel to y-axis, then the value of k is –

Solution : A straight line is parallel to y-axis, if its y-coefficient is zero

i.e. 4 – k = 0   i.e.   k = 4

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Example 4 : If $\lambda x^2 – 10xy + 12y^2 + 5x – 16y – 3$ = 0 represents a pair of straight lines, then $\lambda$ is equal to –

Solution : Comparing with $ax^2+2hxy+by^2+2gx+2fy+c$ = 0

Here a = $\lambda$, b = 12, c = -3, f = -8, g = 5/2, h = -5

Using condition $abc+2fgh-af^2-bg^2-ch^2$ = 0, we have

$\lambda$(12)(-3) + 2(-8)(5/2)(-5) – $\lambda$(64) – 12(25/4) + 3(25) = 0

$\implies$   -36$\lambda$ + 200 – 64$\lambda$ – 75 + 75 = 0

$\implies$   100$\lambda$ = 200

$\therefore$   $\lambda$ = 2

Practice these given straight line examples to test your knowledge on concepts of straight lines.