Let the parabola , meet the coordinate axes at the points P, Q and R. If the circle C with centre at (–1, –1) passes through the points P, Q and R, then the area of PQR is : [2025]

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Solution

Q.
Let the parabola $y = x^{2} + p x - 3$ , meet the coordinate axes at the points P, Q and R. If the circle C with centre at (–1, –1) passes through the points P, Q and R, then the area of $△$ PQR is : [2025]

1 7

2 4

3 6

4 5

Ans.
(3)

Given, equation of parabola is $y = x^{2} + p x - 3$ and equation of circle with centre at (–1, –1) is

${(x + 1)}^{2} + {(y + 1)}^{2} = r^{2}$ ... (i)

Let P( $α$ , 0), Q( $β$ , 0) and R(0, –3) are the three points.

Now, circle passes through point R(0, –3)

$\Rightarrow 1 + {(- 2)}^{2} = r^{2} \Rightarrow 5 = r^{2}$

Put y = 0 in equation (i), we get

${(x + 1)}^{2} + 1 = 5$

$\Rightarrow {(x + 1)}^{2} = 4 \Rightarrow x = 1 o r x = - 3$

$\Rightarrow$ Point are P(1, 0) and (–3, 0)

$∴ Area of △ P Q R = \frac{1}{2}$ $| \begin{matrix} 1 & 0 & 1 \\ - 3 & 0 & 1 \\ 0 & - 3 & 1 \end{matrix} |$ $= 6 sq . units$

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