Question

Obtain the equation of the conic, a focus of which lies at (2,1), the directrix of which is x+y=0 and which passes through (1,4). Also identify the conic.

Answer 1

Answer to Question #91542 – Math – Analytic Geometry

Question

Obtain the equation of the conic, a focus of which lies at $(2,1)$, the directrix of which is $x+y=0$ and which passes through $(1,4)$. Also identify the conic.

Solution

We need to find the distance between $(2,1)$ and $(1,4)$

Using distance formula:

$$\sqrt{(1 - 2)^2 + (4 - 1)^2}$$ $$\sqrt{\frac{(1 + 9)}{\sqrt{10}}}$$

Distance of the point $(1,4)$ from the directrix $x+y=0$ is

$$\frac{(1 + 4)}{\sqrt{2}} = \frac{5}{\sqrt{2}}$$

Ratio of distance is $\frac{\sqrt{10}}{5 / \sqrt{2}}$

This ratio is less than 1, so this is an ellipse.

Its equation is obtained from ratio of the distance of a point on ellipse say $(x,y)$ from focus $(2,1)$ and its distance from the directrix $x+y=0$ is being $2 / \sqrt{5}$.

The latter is $x+y / \sqrt{2}$.

Thus, the equation is

$$\frac{(x - 2)^2 + (y - 1)^2}{\left(\frac{x + y}{\sqrt{2}}\right)^2} = \left(\frac{2}{\sqrt{5}}\right)^2 = \frac{4}{5}$$ $$5 \left(x^2 - 4x + 4 + y^2 - 2y + 1\right) = 2 \left(x^2 + 2xy + y^2\right)$$ $$5x^2 - 20x + 20 + 5y^2 - 10y + 5 = 2x^2 + 4xy + 2y^2$$ $$3x^2 - 4xy - 20x + 3y^2 - 10y + 25 = 0$$

This conic is Ellipse.

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