(iii) Calculate the acute angle between L1 and L3. (b) Given the equation of an ellipse 25x² + 9y² - 150x - 72y + 144 = 0, find the (i) centre, foci and eccentricity, and (ii) the... (iii) Calculate the acute angle between L1 and L3. (b) Given the equation of an ellipse 25x² + 9y² - 150x - 72y + 144 = 0, find the (i) centre, foci and eccentricity, and (ii) the coordinates of the vertices and equations of the directrices. QUESTION THREE (a) Calculate the centre and radius of a circle that passes through the points A(0,2), B(2,2), C(-2,-2). Hence find the equation of the circle. (b) Show that the shortest distance between point P1(x1,y1) and the line Ax + By + c = 0 is given by d = (Ax1 + By1 + c) / √(A² + B²) Read more

Steps to Solve

1. Calculate the acute angle between L1 and L3 To calculate the acute angle between two lines given by their slopes $m_1$ and $m_3$, we can use the formula: $$ \theta = \tan^{-1}\left(\frac{m_1 - m_3}{1 + m_1 m_3}\right) $$ We need the slopes of L1 and L3 to proceed.

2. Find the center, foci, and eccentricity of the ellipse First, we need to rewrite the equation of the ellipse: $$ 25x^2 + 9y^2 - 150x - 72y + 144 = 0 $$ We will complete the square for $x$ and $y$ individually.

   • For $x$, rearrange and complete the square.
   • For $y$, do the same. The center $(h, k)$ will be derived from this form.

3. Identify the standard form of the ellipse After completing the square, the equation should be in the form: $$ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 $$ From this, we can find the foci $(h \pm c, k)$ where $c = \sqrt{a^2 - b^2}$ and the eccentricity $e = \frac{c}{a}$.

4. Find the vertices and directrices The coordinates of the vertices will be $(h \pm a, k)$ and the directrices can be calculated using:

   • For a horizontal ellipse, the equation of the directrices is $x = h \pm \frac{a}{e}$.

5. Calculate the center and radius of the circle The equation of the circle is given by: $$ (x - h)^2 + (y - k)^2 = r^2 $$ To find the center and radius that passes through points A(0,2), B(2,2), C(-2,-2), we will average the coordinates of the points to find the centre $(h, k)$.

6. Determine the equation of the circle After finding the center and radius, substitute back into the circle's equation to obtain the final form.

7. Show the shortest distance formula We will use the distance formula derived from geometry to prove: $$ d = \frac{|Ax_1 + By_1 + c|}{\sqrt{A^2 + B^2}} $$ This involves substituting point P1 into the line equation to derive the shortest distance.