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

Question

If the image of the point \( P(1, 2, a) \) in the line \[ \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{7 - z}{2} \] is \( Q(5, b, c) \), then \( a^2 + b^2 + c^2 \) is equal to

Answer 1

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

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

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

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

Solution and Explanation

Top Questions on Distance of a Point From a Line

Questions Asked in CBSE Class XI exam