First slide

Equation of line in Straight lines

Question

A line passing through the point P(1, 2) meets the line $x + y = 7$ at the distance of 3 units from P. Then the slope of this line satisfies the equation

Moderate

A

$7 x^{2} - 18 x + 7 = 0$
B

$16 x^{2} - 39 x + 16 = 0$
C

$7 x^{2} - 6 x - 7 = 0$
D

$8 x^{2} - 9 x + 1 = 0$

Solution

An equation of line through $(1, 2)$ is $y - 2 = m (x - 1)$ A point on this line is of the form $Q (x, m x - m + 2)$ . This point will lie on the line $x + y = 7$ if
$\begin{array}{l} x + m x - m + 2 = 7 \Rightarrow x = \frac{m + 5}{m + 1} \\ As P Q = 3, P Q^{2} = 9 \\ \Rightarrow (x - 1)^{2} + (m x - m)^{2} = 9 \\ \Rightarrow (x - 1)^{2} 1 + m^{2} = 9 \\ \Rightarrow 16 m^{2} + 16 = 9 m^{2} + 18 m + 9 \\ \Rightarrow 7 m^{2} - 18 m + 7 = 0 \end{array}$
$\Rightarrow$ m satisfies the equation $7 x^{2} - 18 x + 7 = 0$

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