Question

Find values of 'a' if distance between
A(3,2) and B(a +3,4-a) is 2.​

Answer 1

Given :-

The distance between A(3 , 3) and B(a+3 , 4-a) = 2

• x1 = 3

• x2 = a + 3

• y1 = 3

• y2 = 4 - a

Solution :- b

We know that the distance between two points (x1 , y1) and (x2 , y2) is

• $\sqrt{( {x2 - x1)^{2} + (y2 - y1)}^{2} }$

Then,

$: \implies \sqrt{({a + 3 - 3) ^{2}+ (4 - a - 3)}^{2} } = 2 \\ \\ : \implies \sqrt{ {(a)}^{2} + ( {1 - a)}^{2} } = 2 \\ \\ : \implies \sqrt{ {a}^{2} + 1 + {a}^{2} - 2a} = 2 \\ \\ : \implies \sqrt{2 {a}^{2} - 2a + 1} = 2$

Squaring the both sides →

$</strong><strong>: \implies</strong><strong> </strong><strong>( { \sqrt{ {2a}^{2} - 2a + 1 } )}^{2} = ( {2)}^{2}$

$</strong><strong>: \implies</strong><strong> {2a}^{2} - 2a + 1 = 4 \\ \\ </strong><strong>: \implies</strong><strong> </strong><strong>2 {a}^{2} - 2a + 1 - 4 = 0 \\ \\ </strong><strong>: \implies</strong><strong> </strong><strong>2 {a}^{2} - 2a - 3 = 0$

Comparing the equation with ax² + bx + c →

• a = 2

• b = -2

• c = -3

Using the quadratic formula :-

$</strong><strong>: \implies</strong><strong> </strong><strong>a</strong><strong> = \frac{ - b </strong><strong>±</strong><strong> \sqrt{ {b}^{2} - 4ac} }{2a}$

$</strong><strong>: \implies</strong><strong> </strong><strong>a</strong><strong> = \frac{ - ( - 2) </strong><strong>±</strong><strong> \sqrt{( { - 2)}^{2} - 4 \times 2 \times ( - 3)} }{2 \times 2} \\ \\ </strong><strong>: \implies</strong><strong> </strong><strong>a</strong><strong> = \frac{ 2 </strong><strong>±</strong><strong> \sqrt{4 + 24} }{4} \\ \\ a = \frac{2 </strong><strong>±</strong><strong> \sqrt{28} }{4} \\ \\ a = \frac{2 </strong><strong>±</strong><strong>\sqrt{2 \times 2 \times 2 \times 3} }{4} \\ \\ a = \frac{2 </strong><strong>±</strong><strong> 2 \sqrt{6} }{4} \\ \\ a = \frac{2(1 </strong><strong>±</strong><strong> \sqrt{6)} }{4} \\ \\ a = \frac{1 </strong><strong>±</strong><strong> \sqrt{6} }{2}$