Revision: Co-ordinate Geometry JEE Main Co-ordinate Geometry

Locus is the path traced by a moving point, which moves so as to satisfy a certain given condition/conditions. 

The slope m of a line is m = tan⁡θ

where θ is the inclination of the line with the positive x-axis.

x-intercept: Point where a line cuts the x-axis, y = 0

y-intercept: Point where a line cuts the y-axis, x = 0

An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.

The equation of the line with intercepts a and b is \[\frac{x}{a} + \frac{y}{b} = 1\] .Since the line meets the coordinate axes at A and B, the coordinates are A (a, 0) and B (0, b).
Let the given point be P (3, 4).
Here,

\[AP : BP = 2 : 3\]

\[\therefore 3 = \frac{2 \times 0 + 3 \times a}{2 + 3}, 4 = \frac{2 \times b + 3 \times 0}{2 + 3}\]

\[ \Rightarrow 3a = 15, 2b = 20\]

\[ \Rightarrow a = 5, b = 10\]

Hence, the equation of the line is

\[\frac{x}{5} + \frac{y}{10} = 1\]

\[ \Rightarrow 2x + y = 10\]

A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. 

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.

The distance between P(x₁, y₁) and Q(x₂, y₂) is

\[\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]

 The distance of a point P(x, y) from the origin is

\[\sqrt{x^2+y^2}\]

\[P\left(\frac{m_1x_2+m_2x_1}{m_1+m_2},\frac{m_1y_2+m_2y_1}{m_1+m_2}\right)\]

\[m=\frac{y_2-y_1}{x_2-x_1}\]

For line ax + by + c = 0

x-intercept:

\[\left(-\frac{c}{a},0\right)\]

y-intercept:

\[\left(0,-\frac{c}{b}\right)\]

When slope and y-intercept are given

y = mx + c​

• m = slope

• c = y-intercept (value of y when x = 0)

When two points are given

\[\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}\]​​

When the slope and one point are given

y − y₁ = m(x − x₁)

Let P(x, y), Q(a + b, b – a) and R (a – b, a + b) be the given points. Then

PQ = PR   ...(Given)

⇒ `sqrt({x - (a + b)}^2 + {y - (b - a)}^2) = sqrt({x - (a - b)}^2 + {y - (a + b)}^2`

⇒ `{x - (a + b)}^2 + {y - (b - a)}^2 = {x - (a - b)}^2 + {y - (a + b)}^2`

⇒ x² – 2x(a + b) + (a + b)² + y² – 2y(b – a) + (b – a)² = x² + (a – b)² – 2x(a – b) + y² – 2(a + b) + (a + b)²

⇒ –2x(a + b) – 2y(b – a) = –2x(a – b) – 2y(a + b)

⇒ ax + bx + by – ay = ax – bx + ay + by

⇒ 2bx = 2ay

⇒ bx = ay

As the point R divides the line segment AB externally, we have either A-B-R or R-A-B.

Assume that A-B-R and `bar(AR) : bar(BR)` = m : n

∴ `(AR)/(BR) = m/n` so n(AR) = m(BR) 

As `n(bar(AR))` and `m(bar(BR))` have same magnitude and direction,

∴ `n(bar(AR)) = m(bar(BR))`

∴ `n(barr - bara) = m(barr - barb)`

∴ `nbarr - nbara = mbarr - mbarb`

∴ `mbarr - nbarr = mbarb - nbara`

∴ `(m - n)barr = mbarb - nbara`

∴ `barr = (mbarb - nbara)/(m - n)`

Hence proved.

R is a point on the line segment AB(A – R – B) and `bar(AR)` and `bar(RB)` are in the same direction.

Point R divides AB internally in the ratio m : n

∴ `(AR)/(RB) = m/n`

∴ n(AR) = m(RB)

As `n(bar(AR))` and `m(bar(RB))` have same direction and magnitude,

`n(bar(AR)) = m(bar(RB))`

∴ `n(bar(OR) - bar(OA)) = m(bar(OB) - bar(OR))`

∴ `n(vecr - veca) = m(vecb - vecr)`

∴ `nvecr - nveca = mvecb - mvecr`

∴ `mvecr + nvecr = mvecb + nveca`

∴ `(m + n)vecr = mvecb + nveca`

∴ `vecr = (mvecb + nveca)/(m + n)`

Let A, B and C be vertices of a triangle.

Let D, E and F be the mid-points of the sides BC, AC and AB respectively.

Let `bara, barb, barc, bard, bare` and `barf` be position vectors of points A, B, C, D, E and F respectively.

Therefore, by mid-point formula,

∴ `bard = (barb + barc)/2, bare = (bara + barc)/2` and `barf = (bara + barb)/2`

∴ `2bard = barb + barc, 2bare = bara + barc` and `2barf = bara + barb`

∴ `2bard + bara = bara + barb + barc`, similarly `2bare + barb = 2barf + barc = bara + barb + barc`

∴ `(2bard + bara)/3 = (2bare + barb)/3 = (2barf + barc)/3 = (bara + barb + barc)/3 = barg`  ...(Say)

Then we have `barg = (bara + barb + barc)/3 = ((2)bard + (1)bara)/(2 + 1) = ((2)bare + (1)barb)/(2 + 1) = ((2)barf + (1)barc)/(2 + 1)`

If G is the point whose position vector is `barg`, then from the above equation it is clear that the point G lies on the medians AD, BE, CF and it divides each of the medians AD, BE, CF internally in the ratio 2 : 1.

Therefore, three medians are concurrent.

Let `bara, barb, barc, bard, bare, barf` be the position vectors of the points A, B, C, D, E, F respectively.

Since D, E, F are the midpoints of BC, CA, AB respectively, by the midpoint formula

`bard = (barb + barc)/2, bare = (barc + bara)/2, barf = (bara + barb)/2`

∴ `bar(AD) + bar(BE) + bar(CF) = (bard - bara) + (bare - barb) + (barf - barc)`

= `((barb + barc)/2 - bara) + ((barc + bara)/2 - barb) + ((bara + barb)/2 - barc)`

= `1/2barb + 1/2barc - bara + 1/2barc + 1/2bara - barb + 1/2bara + 1/2barb - barc`

= `1/2(barb + barc - 2bara + bar c + bara - 2barb + bara + barb - 2barc)`

= `(bara + barb + barc) - (bara + barb + barc) = bar0`. 

Nature of Slope

• m > 0 → rising line

• m < 0 → falling line

• m = 0 → horizontal line

• m = ∞→ vertical line

Parallel Lines 

Two lines are parallel ⇔ , their slopes are equal, m₁ = m₂

​Perpendicular Lines

Two lines are perpendicular ⇔ m₁ × m₂ = −1

​Collinearity of Three Points

Points A, B, and C are collinear

Method 1: Distance method

AB + BC = AC

Method 2: Slope method

Slope of AB = Slope of BC

• x-intercept:
  Right of origin → positive
  Left of origin → negative

• y-intercept:
  Above origin → positive
  Below origin → negative