Mathematics STANDARD PROPERTIES

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STANDARD PROPERTIES

Position of a Point With Respect to a Parabola `y^2 = 4ax` :

Let `P(x_1, y_1)` be a point. From `P` draw `PM bot AX` (on the axis of parabola) meeting the parabola `y^2 = 4ax`
at `Q (x_1,y_2)` where `Q (x_1, y_2)` lie on the parabola therefore

`y_2^2=4ax_1` ...........................(1)

Now, `P` will be outside, on or inside the parabola

`y^2 = 4ax` acoording as

`PM > , = ,` or `< QM`

`=> (PM)^2> , = , ` or `< (QM)^2`

`=> y_1^2 > ,=, ` or `< y_2^2`

`=> y_1^2 > , = , ` or `<4ax_1` (from (1))

Hence `y_1^2 - 4ax_1 > , = ,` or `< 0`

Hence in short, equation of parabola `S(x, y) = y^2 - 4ax`.

(i) If `S(x_1, y_1) > 0` then `P(x_1 ,y_1)` lie outside the parabola.

(ii) If `S(x_1, y_1) < 0` then `P(x_1 ,y_1)` lie inside the parabola.

(iii) If `S(x_1, y_1) = 0` then `P(x_1, y_1)` lie on the parabola.

This result holds true for circle, parabola and ellipse.

Let `P(x_1, y_1)` be a point. From `P` draw `PM bot AX` (on the axis of parabola) meeting the parabola `y^2 = 4ax`
at `Q (x_1,y_2)` where `Q (x_1, y_2)` lie on the parabola therefore

`y_2^2=4ax_1` ...........................(1)

Now, `P` will be outside, on or inside the parabola

`y^2 = 4ax` acoording as

`PM > , = ,` or `< QM`

`=> (PM)^2> , = , ` or `< (QM)^2`

`=> y_1^2 > ,=, ` or `< y_2^2`

`=> y_1^2 > , = , ` or `<4ax_1` (from (1))

Hence `y_1^2 - 4ax_1 > , = ,` or `< 0`

Hence in short, equation of parabola `S(x, y) = y^2 - 4ax`.

(i) If `S(x_1, y_1) > 0` then `P(x_1 ,y_1)` lie outside the parabola.

(ii) If `S(x_1, y_1) < 0` then `P(x_1 ,y_1)` lie inside the parabola.

(iii) If `S(x_1, y_1) = 0` then `P(x_1, y_1)` lie on the parabola.

This result holds true for circle, parabola and ellipse.

Interaction Between The Line And Parabola :

Let the parabola be `y^2 = 4ax` .....................(i)

and the given line be `y = mx + c` .............(ii)

then line may cut, touch or does not meet parabola.

The points of intersection of the line `( 1)` and the parabola `(2)` will be obtained by solving the two equations simultaneously. By solving equation `(i)` and `(ii)`, we get

`my^2 - 4ay + 4ac = 0`

this equation has two roots and its nature will be decided by the discriminent `D = 16a(a - cm)`

Now, if `D > 0` i. e., `c < a/m` , then line intersect the parabola at two distinct points.

lf `D = 0` i.e., `c =a/m` , then line touches the parabola . (It is condition of tangency)

If `D < 0` i.e. , `c > a/m`, then line neither touch nor intersect the parabola.

Let the parabola be `y^2 = 4ax` .....................(i)

and the given line be `y = mx + c` .............(ii)

then line may cut, touch or does not meet parabola.

The points of intersection of the line `( 1)` and the parabola `(2)` will be obtained by solving the two equations simultaneously. By solving equation `(i)` and `(ii)`, we get

`my^2 - 4ay + 4ac = 0`

this equation has two roots and its nature will be decided by the discriminent `D = 16a(a - cm)`

Now, if `D > 0` i. e., `c < a/m` , then line intersect the parabola at two distinct points.

lf `D = 0` i.e., `c =a/m` , then line touches the parabola . (It is condition of tangency)

If `D < 0` i.e. , `c > a/m`, then line neither touch nor intersect the parabola.

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