Loading [MathJax]/jax/output/HTML-CSS/jax.js

Subscribe to our YouTube channel for the latest videos, updates, and tips.


Position of a Point with respect to a Parabola

We will learn how to find the position of a point with respect to a parabola.

The position of a point (x1, y1) with respect to a parabola y2 = 4ax (i.e. the point lies outside, on or within the parabola) according as y12 - 4ax1 >, =, or < 0.



Let P(x1, y1) be a point on the plane. From P draw PN perpendicular to the x-axis i.e., AX and N being the foot of the perpendicular.

PN intersect the parabola y2 = 4ax at Q and let the coordinates of Q be (x1, y2). Now, the point Q (x1, y2) lies on the parabola y2 = 4ax. Hence we get

y22 = 4ax1

Therefore, the point


(i) P lies outside the parabola y2 = 4ax if PN > QN

i.e., PN2 > QN2

y12 > y22

y12 > 4ax1, [Since, 4ax1 = y22].


(ii) P lies on the parabola y2 = 4ax if PN = QN

i.e., PN2 = QN2

y12 = y22

y12 = 4ax1, [Since, 4ax1 = y22].


(iii) P lies outside the parabola y2 = 4ax if PN < QN

i.e., PN2 < QN2

y12 < y22

y12 < 4ax1, [Since, 4ax1 = y22].

Therefore, the point P (x1, y1) lies outside, on or within the parabola y2 = 4ax according as

y12 - 4ax1 >,= or < 0.


Notes:

(i) The point P(x1, y1) lies outside, on or within the parabola y2 = -4ax according as y12 + 4ax1 >, = or <0.

(ii) The point P(x1, y1) lies outside, on or within the parabola x2 = 4ay according as x12 - 4ay1 >, = or <0.

(ii) The point P(x1, y1) lies outside, on or within the parabola x2 = -4ay according as x12 + 4ay1 >, = or <0.

Solved examples to find the position of the point P (x1, y1) with respect to the parabola y2 =  4ax:

1. Does the point (-1, -5) lies outside, on or within the parabola y2 = 8x?

Solution:

We know that the point (x1, y1) lies outside, on or within the parabola y2 = 4ax according as y12 - 4ax1 is positive, zero or negative.

Now, the equation of the given parabola is y2 = 8x ⇒ y2 - 8x= 0

Here x1 = -1 and y1 = -5

Now, y12 - 8x1  = (-5)2 - 8 ∙ (-1) = 25 + 8 = 33 > 0

Therefore, the given point lies outside the given parabola.

 

2. Examine with reasons the validity of the following statement:

"The point (2, 3) lies outside the parabola y2 = 12x but the point (- 2, - 3) lies within it."

Solution:         

We know that the point (x1, y1) lies outside, on or within the parabola y2 = 4ax according as y12 - 4ax1 is positive, zero or negative.

Now, the equation of the given parabola is y2 = 12x or, y2 - 12x = 0

For then point (2, 3):

Here x1 = 2 and y1 = 3

Now, y12 - 12x1 = 32 – 12 ∙ 2 = 9 - 24 = -15 < 0

Hence, the point (2, 3) lies within the parabola y2 = 12x.

For then point (-2, -3):

Here x1 = -2 and y1 = -3

Now, y12 - 12x1 = (-3)2 – 12 ∙ (-2) = 9 + 24 = 33 > 0

Hence, the point (-2, -3) lies outside the parabola y2 = 12x.

Therefore, the given statement is not valid.

● The Parabola




11 and 12 Grade Math 

From Position of a Point with respect to a Parabola to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Average | Word Problem on Average | Questions on Average

    May 19, 25 02:53 PM

    Worksheet on Average
    In worksheet on average we will solve different types of questions on the concept of average, calculating the average of the given quantities and application of average in different problems.

    Read More

  2. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

    May 18, 25 04:33 PM

    Printable eight times table
    In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

    Read More

  3. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  4. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More

  5. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 16, 25 11:13 AM

    Round off to Nearest One
    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More