First slide

Definition of a circle

Question

If the base of a triangle and the ratio of the lengths of the other two unequal sides are given, then the vertex lies on $a / a n$

Moderate

A

straight line
B

circle
C

ellipse
D

parabola

Solution

Let the base of the triangle be the line segment joining $B (0, 0)$ and $C (a, 0)$ and let the vertex be $A (h, k),$ where a is fixed. Also, let
$\begin{array}{l} \frac{A B}{A C} = λ, λ \neq 1 \\ \Rightarrow A B^{2} = λ^{2} A C^{2} \\ \Rightarrow h^{2} + k^{2} = λ^{2} \{(h - a)^{2} + k^{2}\} \\ \Rightarrow h^{2} (1 - λ^{2}) + k^{2} (1 - λ^{2}) + 2 a λ^{2} h - a^{2} λ^{2} = 0 \end{array}$
So, $A (h, k)$ lies on
$x^{2} + y^{2} + 2 a \frac{λ^{2}}{1 - λ^{2}} x - \frac{a^{2} λ^{2}}{1 - λ^{2}} = 0$ which is a circle

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