Name: Condition for Tangency to a Circle
Uploaded: 2019-07-19
Duration: 3 min
Description: The line y=mx+c touches the circle is called a tangent. How would you find a condition for which line to be a tangent to the circle and its point of contact?

Question

Find the center, the lengths, the equations of the major and minor axes, the length and the equations of latus rectum, foci and directrices of the ellipse

25 x^{2} + 9 y^{2} - 150 x - 90 y + 225 = 0

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Accepted Answer

Given ellipse $25 x^{2} + 9 y^{2} - 150 x - 90 y + 225 = 0$

$25 (x^{2} - 6 x) + 9 (y^{2} - 10 y) = - 225$

$25 (x^{2} - 6 x + 9) + 9 (y^{2} - 10 y + 25) = - 225 + 225 + 225$

$25 {(x - 3)}^{2} + 9 {(y - 5)}^{2} = 225$

$\frac{{(x - 3)}^{2}}{9} + \frac{{(y - 5)}^{2}}{25} = 1$ ......(i)

Shifting the origin to the point $(3, 5)$ we have $(x - 3) = X, (y - 5) = Y$ ....(ii)

Substituting for $x and y in (i)$ , we have $\frac{X^{2}}{9} + \frac{Y^{2}}{25} = 1$ ....(iii)

So, the centre of $\frac{X^{2}}{9} + \frac{Y^{2}}{25} = 1$ is $(0, 0) i . e X = 0, Y = 0$

Putting $X and Y$ in (ii), we get $x = 3, y = 5$

Therefore, centre of (i) is $(3, 5)$

Now, lengths of the semi-major and semi-minor axes are $5 and 3$ respectively

Therefore, lengths of major and minor axes are $10 and 6$

The equations of the major and minor axes of (iii) are $X = 0, Y = 0$ or using (ii), they are $x = 3, y = 5$

Length of the latus rectum are $= \frac{2 b^{2}}{a} = \frac{2 \times 9}{5} = \frac{18}{5}$

Eccentricity $(e) = \sqrt{1 - \frac{9}{25}} = \frac{4}{5}$

The equation of latus recta of (iii) are $Y = \pm a e = \pm 5 \times \frac{4}{5} = \pm 4$

Using (ii), $y - 5 = \pm 4 i . e y = 9 and y = 1$

Coordinates of foci are $(0, \pm a e) = (0, \pm 5 \times \frac{4}{5}) = (0, \pm 4)$

So, $x - 3 = 0, y - 5 = \pm 9$ . Therefore coordinates are $(3, 9) and (3, 1)$

Now, equations of diretrices are $Y = \pm \frac{25}{4} i . e y - 5 = \pm \frac{25}{4}$

$y = 5 \pm \frac{25}{4}$

Therefore, equations are $4 y - 45 = 0 and 4 y + 5 = 0$

Find the center, the lengths, the equations of the major and minor axes, the length and the equations of latus rectum, foci and directrices of the ellipse $25 x^{2} + 9 y^{2} - 150 x - 90 y + 225 = 0$ .

Important Questions on Conic Sections

Learn and Prepare for any exam you want !

Find the center, the lengths, the equations of the major and minor axes, the length and the equations of latus rectum, foci and directrices of the ellipse 25x2+9y2-150x-90y+225=0.

Important Questions on Conic Sections

Learn and Prepare for any exam you want !

Find the center, the lengths, the equations of the major and minor axes, the length and the equations of latus rectum, foci and directrices of the ellipse $25 x^{2} + 9 y^{2} - 150 x - 90 y + 225 = 0$ .