If the normal at $(ap^2, 2ap)$ on the parabola $y^2 = 4ax,$ meets the parabola again at $(aq^2, 2aq)$, then
- $p^2 + pq + 2 = 0 $
- $p^2 - pq + 2 = 0$
- $q^2 + pq + 2 = 0$
- $p^2 + pq + 1 = 0$
The Correct Option is A
Solution and Explanation
$\Rightarrow \; paq^2 + 2aq = 2ap + ap^3$ $\Rightarrow \; p(q^2-p^2) = 2(p - q)$ $\Rightarrow \; p (q + p) = -2$ $\Rightarrow \; p^2 + pq + 2 = 0$
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Concepts Used:
Arithmetic Progression
Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.
For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.
In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.
For eg:- 4,6,8,10,12,14,16
We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.
Read More: Sum of First N Terms of an AP