Show that the line $15 x - 8 y + 44 = 0$ touches the ellipse given by $25 x^{2} + 4 y^{2} - 100 x - 24 y + 36 = 0$ . Find the point of contact.

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Solution

$25 x^{2} + 4 y^{2} - 100 x - 24 y + 36 = 0 ⟹ \frac{(5 x - 10)^{2}}{100} + \frac{(2 y - 6)^{2}}{100} = 0 ⟹ \frac{(x - 2)^{2}}{4} + \frac{(y - 3)^{2}}{25} = 0$

Equation of line $15 x - 8 y + 44 = 0 ⟹ y = \frac{15}{8} x + \frac{11}{2}$
Let the origin be shifted to $(2, 3)$
Then the equations are $\frac{x^{2}}{4} + \frac{y^{2}}{25} = 0, y = \frac{15}{8} x + \frac{19}{4}$

Condition of a line to be tangent to ellipse is $c^{2} = a^{2} m^{2} + b^{2}$
$c^{2} = \frac{361}{16}$
$a^{2} m^{2} + b^{2} = 4 \frac{225}{64} + 25 = \frac{625}{16}$
So, the line is nnot tangent to the equation

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