First slide

Ellipse

Question

$Consider set S = \{(a, b) : (a + 3) t^{2} = 3 - a and {bt}^{2} - 4 t + b = 0\}, where t is a real$
$parameter. Let C be a curve which is formed by all elements of set S where (a, b) is$
$a point in R^{2} . Tangents are drawn from the point P (3, 4) to the curve C touching$
$the curve C at point Q and R . If the circumcentre of triangle PQR is (α, β), then$ $the value of α + 3 β is$

Moderate

Solution

$\begin{array}{l} t^{2} = \frac{3 - a}{3 + a} = \frac{4 t - b}{b} \\ \Rightarrow t = \frac{3 b}{2 (3 + a)} put in eq. (1) \\ \Rightarrow 4 a^{2} + 9 b^{2} = 36 \\ \Rightarrow \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \end{array}$
Equation of tangent from P
$\begin{array}{l} y = mx \pm \sqrt{9 m^{2} + 4} \\ (4 - 3 m) = \pm \sqrt{9 m^{2} + 4} \\ \Rightarrow m = \infty, m = \frac{1}{2} \\ x = 3, x - 2 y + 5 = 0 \\ point of contact Q (3, 0) \end{array}$
$R (- \frac{9}{8}, \frac{8}{5})$
$perpendicular bisector of PQ : y = 2$
$perpendicular bisector of RQ : y - \frac{4}{5} = 3 (x - \frac{3}{5}) .$
circumcentre (1, 2)
$\Rightarrow α + 3 β = 7$

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