Question

Find the equation of the chord joining the points $P(\frac{\pi }{4})\text{and}Q(\frac{\pi }{4})$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1.$

• A. $3x+5y=15\sqrt{2}$
• B. $3x-5y=15\sqrt{2}$
• C. $5x+3y=15\sqrt{2}$
• D. $5x-3y=15\sqrt{2}$

Answer 1

The correct option is A $3x+5y=15\sqrt{2}$

Given: Points $P(\frac{\pi }{4})\text{and}Q(\frac{\pi }{4})$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$

To Find: Equation of chord $PQ$

Step-$1:$ Use the formula for the equation of the chord.

Step-$2:$ Substitute the values to find the equation.

The equation of the chord $PQ$ on the ellipse is
$\frac{x}{a}\cos \frac{\alpha +\beta }{2}+\frac{y}{b}\sin \frac{\alpha +\beta }{2}=\cos \frac{\alpha -\beta }{2}$

$\alpha =\frac{\pi }{4}$, $\beta =\frac{\pi }{4}$

$\Rightarrow \frac{\alpha +\beta }{2}=\frac{1}{2}(\frac{\pi }{4}+\frac{\pi }{4})=\frac{\pi }{4}$

Also, $\frac{\alpha -\beta }{2}=\frac{1}{2}(\frac{\pi }{4}-\frac{\pi }{4})=0$

Equation of chord is $\frac{x}{5}\cos \frac{\pi }{4}+\frac{y}{3}\sin \frac{\pi }{4}=\cos 0$

$\Rightarrow \frac{x}{5}(\frac{1}{\sqrt{2}})+\frac{y}{3}(\frac{1}{\sqrt{2}})=1$

$\Rightarrow 3x+5y=15\sqrt{2}$