Find the equation of parabola whise focus is (2,5) and directrix is 3x+4y+1=0???? ( please write the answer ona paper and send the answer).
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Parabola : F(2,5)
Directrix D = 3x + 4y + 1 = 0
Let a point P(h,k) be on the Parabola.
By definition, parabola is the locus of a point P whose ⊥ distance from the directrix D is same as the distance from Focus F.
PF² = (h-2)² + (k-5)²
(Distance of P from D)² = (3 h + 4k + 1)² / (3²+4²)
Hence, [h²-4h+4+ k²-10k+25] *25 = 9h²+16k²+1+24hk+8k+6h
16h² + 9k² -106 h -258 k - 24 h k + 724 = 0
Replace (h,k) by (x,y)
Parabola: 16 x² + 9 y² - 106 x - 258 y - 24 x y + 724 = 0
Rewrite it as:
(4x - 3y + w)² = w²+ 8w x - 6 wy + 106 x + 258 y - 724
= x(8w+106) + y(258-6w) + (w²-724)
Finding w from (258 - 6w)/(8w+106) = 4/3, we get: w = 7
So equation of parabola : (4x- 3y +7)² = 27 (6x + 8y - 25)
Vertex at : (19/50, 71/25). Axis: 6x+8y - 25 = 0
Directrix D = 3x + 4y + 1 = 0
Let a point P(h,k) be on the Parabola.
By definition, parabola is the locus of a point P whose ⊥ distance from the directrix D is same as the distance from Focus F.
PF² = (h-2)² + (k-5)²
(Distance of P from D)² = (3 h + 4k + 1)² / (3²+4²)
Hence, [h²-4h+4+ k²-10k+25] *25 = 9h²+16k²+1+24hk+8k+6h
16h² + 9k² -106 h -258 k - 24 h k + 724 = 0
Replace (h,k) by (x,y)
Parabola: 16 x² + 9 y² - 106 x - 258 y - 24 x y + 724 = 0
Rewrite it as:
(4x - 3y + w)² = w²+ 8w x - 6 wy + 106 x + 258 y - 724
= x(8w+106) + y(258-6w) + (w²-724)
Finding w from (258 - 6w)/(8w+106) = 4/3, we get: w = 7
So equation of parabola : (4x- 3y +7)² = 27 (6x + 8y - 25)
Vertex at : (19/50, 71/25). Axis: 6x+8y - 25 = 0
kvnmurty:
click on red heart thanks above pls
Answered by
6
given ,
focus (2 ,5)
and directrix 3x + 4y + 1 =0
we know ,
a parobala is the locus of a point which moves in a plane such that its distance from a fixed point is equal to its distance from fixed line. (directrix)
e.g let a point ( r , s )
then by concept of parabola ,
| 3r + 4s + 1|/5 = {(r -2)^2 + (s -5)^2}^1/2
take square both side ,
(3r + 4s + 1)^2/25 = (r -2)^2 + (s -5)^2
9r^2 + 16s^2 + 1 +24rs + 8s +6r = 25r ^2 +25s^2 +625 + 100 -100r - 250s
16r ^2 + 9s^2 -24rs -106r -258s +724 =0
now ,
put. r = x. and s = y
16x ^2 + 9y^2 -24xy -106x -258y +724 =0
focus (2 ,5)
and directrix 3x + 4y + 1 =0
we know ,
a parobala is the locus of a point which moves in a plane such that its distance from a fixed point is equal to its distance from fixed line. (directrix)
e.g let a point ( r , s )
then by concept of parabola ,
| 3r + 4s + 1|/5 = {(r -2)^2 + (s -5)^2}^1/2
take square both side ,
(3r + 4s + 1)^2/25 = (r -2)^2 + (s -5)^2
9r^2 + 16s^2 + 1 +24rs + 8s +6r = 25r ^2 +25s^2 +625 + 100 -100r - 250s
16r ^2 + 9s^2 -24rs -106r -258s +724 =0
now ,
put. r = x. and s = y
16x ^2 + 9y^2 -24xy -106x -258y +724 =0
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