First slide

Equation of line in Straight lines

Question

If the equal sides AB and AC (each equal to a) of a right angled isosceles triangle ABC be produced to P and Q so that $B P \times C Q = A B^{2}$ , then the line PQ always passes through the fixed point

Moderate

A

(a, 0)
B

(0, a)
C

(a, a)
D

none of these

Solution

We take A as the origin and AB and AC as x-axis and y-axis respectively.
Let AP = h, AQ = k.
Equation of the line PQ is
$\frac{x}{h} + \frac{y}{k} = 1$ ------(1)
Given, $B P \times C Q = A B^{2}$
$\Rightarrow (h - a) (k - a) = a^{2}$
$\Rightarrow h k - a k - a h + a^{2} = a^{2}$ or ak + ha = hk
or $\frac{a}{h} + \frac{a}{k} = 1$ ------(2)
From (2), it follows that line (1) i.e., PQ passes through the fixed point (a, a)

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