Question

If the lines p(p² + 1)x - y + q = 0 and (p2 + 1)²x + (p2 + 1)y + 2q = 0 are perpendicula to the same line, then find the vaue of p will be,
1. 1
2. -1
3. 0
4. -2

Answer 1

Correct Answer - Option 2 : -1

Concept:

The general equation of a line is y = mx + c   -----(A)

Where m is the slope and c is any constant.

• The slope of parallel lines is equal.
• Slope of the perpendicular line have their product = -1

Calculation:

The given lines are, p(p2 + 1)x - y + q = 0    ------(i)

and (p2 + 1)2x + (p2 + 1)y + 2q = 0               -------(ii)

Slope of the lines (i) is, \(\frac{-p(p^2+1)}{-1}\) i.e, p(p² + 1)

Similarly slope of the line (ii) is, \(-\frac{(p^2+1)^2}{p^2+1}\) i.e, -(p² +1)

Since the given lines are perpendicular to the same line, therefore, these lines are parallel which gives the lines have equal slope.

⇒ p(p2 + 1) = -(p2 +1)

⇒ (p² + 1) (p + 1) = 0

but p² + 1 ≠ 0

⇒ p + 1 = 0

⇒ p = -1