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Mathematics Question on Distance of a Point From a Line

The line through the points (h, 3) and (4, 1) intersects the line 7x−9y−19=07x - 9y - 19 = 0. at right angle. Find the value of h.

Answer

The slope of the line passing through points (h, 3) and (4, 1) is m1=(1−3)(4−h)=−2(4−h)m_1 = \frac{(1-3)}{(4-h)} = \frac{-2}{(4-h)}
The slope of line 7x\-9y\-19=07x \- 9y \- 19 = 0 or y=79x–199y = \frac{7}{9}x – \frac{19}{9} is m2=79m_2 =\frac{ 7}{9}
It is given that the two lines are perpendicular.
∴m1×m2=−1∴m_1 × m_2 = -1

⇒−2(4−h)×79=−1⇒\frac{-2}{(4-h)} \times\frac{ 7}{9} = -1

⇒−14(36−9h)=−1⇒\frac{-14}{\left(36-9h\right)} = -1
⇒−14=−1×(36–9h)⇒-14= -1\times(36 – 9h)
⇒36–9h=14⇒36 – 9h = 14
⇒9h=36–14⇒9h = 36 – 14
⇒h=229⇒h =\frac{ 22}{9}
Thus, the value of h is 229.\frac{22}{9}.