Quadratic equations - Area!

Geometry Level 3

The area of region in $xy$ -plane consisting of all points $(a,b)$ such that quadratic equation $ax^2 + 2(a + b - 7)x + 2b = 0$ has fewer than $2$ real solutions is?

4 \pi

25 \pi

9 \pi

49 \pi

1 solution

Satvik Choudhary
Jun 10, 2015

Less than two solutions implies Discriminant is not greater than 0.

Note: I am using less than sign instead of less than equal to sign.

$(2 (a+b-7))^{2}-4.2b.a <0$ Dividing by 4 $a^{2}+b^{2}+7^{2}+2ab-14a-14b-2ab <0$

$(a-7)^{2}+(b-7)^{2}<7^{2}$

This is the equation of a circle having radius 7. So area is $\boxed{49\pi}$ .

Quadratic equations - Area!

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