Question

What is an equation of conic with the property that the distance of its point from points (2,0) is twice as much as its distance from the line 2x-1=0

Answer 1

SOLUTION

TO DETERMINE

The equation of conic with the property that the distance of its point from points (2,0) is twice as much as its distance from the line 2x - 1 = 0

EVALUATION

Let P(x, y) be the point

Then the distance of the point P from the point (2,0) is

$= \sf{ \sqrt{ {( x - 2)}^{2} + {(y - 0)}^{2} } }$

Again the distance of the point P from the line 2x - 1 = 0 is

$\displaystyle \sf{ = \bigg| \frac{2x - 1}{ \sqrt{ {2}^{2} + {0}^{2} } } \bigg| }$

$\displaystyle \sf{ = \bigg| x - \frac{1}{2} \bigg| }$

So by the given condition

$\displaystyle \sf{ \sqrt{ {(x - 2)}^{2} + {(y - 0)}^{2} } = 2 \times \bigg| x - \frac{1}{2} \bigg| }$

$\displaystyle \sf{ \implies \sqrt{ {(x - 2)}^{2} + {(y - 0)}^{2} } = \bigg| 2 x - 1 \bigg| }$

Squaring both sides we get

$\displaystyle \sf{ \implies {(x - 2)}^{2} + {(y - 0)}^{2} = {(2x - 1)}^{2} }$

$\displaystyle \sf{ \implies {x}^{2} - 4x + 4 + {y}^{2} = 4 {x}^{2} - 4x + 1 }$

$\displaystyle \sf{ \implies 3 {x}^{2} - {y}^{2} = 3 }$

Hence the required equation of the conic is

$\displaystyle \sf{ 3 {x}^{2} - {y}^{2} = 3 }$

The above conic represents an equation of a Hyperbola

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. A hyperbola has its center at (3, 4), a vertex at the point (9, 4), and the length of its latus rectum is 3 units.

https://brainly.in/question/30210645

2. The length of the latus rectum of the parabola

13[(x-3)^2+(y-4)^2 )= (2x-3y+ 5)^2 is

https://brainly.in/question/24980361