Question

the directrix of a conic section is the straight line 3x-4y+5=0 and the focus is (2,3). if the eccentricity e is 1, find the equation to the conic section. is the conic section a parabola?​

Answer 1

Step-by-step explanation:

We know that

SP=e PM

So let the points in the conic be (x, Y)

(3X-4Y+5)/25 = (1) sqrt((x-2)^2 +(y-3)^2)

Answer 2

Answer:

The equation of conic section is 16 x² - 9 y² - 40 x  - 185 y + 300 = 0

Step-by-step explanation:

Given as :

The equation of directrix of conic section is 3 x - 4 y + 5 = 0

The co-ordinate of focus = s = 2 , 3

The eccentricity = e = 1

Let any point be m

According to question

Let x , y be the point at point p

So, sp = e pm

Now, The distance between sp = $\sqrt{(x - 2)^{2} + (y - 3)^{2} }$

The measure of pm = $\dfrac{3x - 4 y + 5}{\sqrt{3^{2} + 4^{2} } }$

Or, pm = $\dfrac{3 x - 4 y +5}{5}$

∵ sp = e pm

Or, $\sqrt{(x - 2)^{2} + (y - 3)^{2} }$ = 1 × $\dfrac{3x - 4 y + 5}{\sqrt{3^{2} + 4^{2} } }$

Or, $\sqrt{(x - 2)^{2} + (y - 3)^{2} }$ =  $\dfrac{3x - 4 y + 5}{5 }$

Or, Squaring both side

(x - 2)² + (y - 3)² = [ $\dfrac{3x - 4 y + 5}{5 }$] ²

Or, 25 [ (x - 2)² + (y - 3)² ] = (3 x - 4 y + 5)²

Or, 25 [ x² - 4 x + 4 + y² - 9 y + 9 ] = (3 x + 5)² + 16 y² + 2 × 5 × (3 x - 4 y)

Or, 25 x² - 100 x + 25 y² - 225 y + 325 = 9 x² + 30 x + 25 + 16 y² + 30 x - 40y

Or , 16 x² - 9 y² - 40 x  - 185 y + 300 = 0

Hence, The equation of conic section is 16 x² - 9 y² - 40 x  - 185 y + 300 = 0 Answer