Question

If (0,1) is equidistant from p(5,-3) and r(x, 6) find the value of x also find the distance qr and pr

Answer 1

EXPLANATION.

If ( 0,1) is equidistant from P ( 5,-3) and R ( x,6).

To find the value of x also find the distance of QR and PR.

Let x₁ = 0  and  y₁ = 1

Let x₂ = 5  and  y₂ = -3.

By using the distance formula,

QP = √( x₁ - x₂)² + ( y₁ - y₂)².

QP = √( 0 - 5 )² + ( 1 - (-3))².

QP = √(-5)² + (4)².

QP = √25 + 16.

QP = √41.

Let x₁ = 0  and  y₁ = 1

Let x₂ = x  and  y₂ = 6.

By using the distance formula,

√(x₁ - x₂)² + (y₁ - y₂)².

QR = √(0 - x )² + (1 - 6 )².

QR = √(-x)² + (-5)².

QR = √x² + 25.

It is equidistance form the point P and Q.

QP = QR.

√41 = √x² + 25.

41 = x² + 25.

41 - 25 = x².

16 = x².

x = √16.

x = ± 4.

Point of R ( 4,6 )  and ( -4,6).

QR = √x² + 25.

Put QR = 4 in equation.

QR = √(4)² + 25.

QR = √41.

Put QR = -4 in equation,

QR = √x² + 25.

QR = √(-4)² + 25.

QR = √41.

To find PR.

P ( 5,-3)  and  R (x,6).

Let =  x₁ = 5  and  y₁ = -3.

Let = x₂ = x  and  y₂ = 6.

By using the distance formula,

PR = √(x₁ - x₂)² + (y₁ - y₂)².

PR = √(5 - x)² + (-3 - 6)².

PR = √(5 - x)² + (9)².

Put x = 4 in equation.

PR = √( 5 - 4)² + (9)².

PR = √1 + 81.

PR =  √82.

Put x = -4 in equation,

PR = √(5 -(-4))² + (9)².

PR = √(9)² + (9)².

PR = √81 + 81.

PR = √162.

Answer 2

$\Large\bf{\color{cyan}GiVeN,}$

• Q(0 , 1) is equidistant from P(5 , -3) and R(x , 6).

$\Large\bf{\color{coral}CaLcUlAtIoN,}$

$\bf\pink{According\:to\:the\:question,}$

$\longmapsto\:\:\bf\blue{QP\:=\:QR}$

➡ Squaring both sides,

$\longmapsto\:\:\bf{(QP)^2\:=\:(QR)^2}$

$\bf\pink{We\:know\:that,}$

$\orange\bigstar\:\:\bf\green{Distance\:formula\:=\:\sqrt{(x_2\:-\:x_1)^2\:+\:(y_2\:-\:y_1)^2}\:}$

$\bf\red{So,}$

$\longmapsto\:\:\bf{\Big(\sqrt{(5\:-\:0)^2\:+\:(-3\:-\:1)^2}\Big)^2\:=\:\Big(\sqrt{(x\:-\:0)^2\:+\:(6\:-\:1)^2}\Big)}$

$\longmapsto\:\:\bf{5^2\:+\:(-4)^2\:=\:x^2\:+\:5^2}$

$\longmapsto\:\:\bf{x^2\:=\:16}$

$\longmapsto\:\:\bf{x\:=\:\sqrt{16}}$

$\longmapsto\:\:\bf\purple{x\:=\:\pm\:4}$

FOR QR ;-

$\red\checkmark\:\:\bf{QR\:=\:\sqrt{(\pm{4}\:-\:0)^2\:+\:(6\:-\:1)^2}\:}$

$:\implies\:\:\bf{QR\:=\:\sqrt{(\pm{4})^2\:+\:5^2}\:}$

$:\implies\:\:\bf{QR\:=\:\sqrt{16\:+\:25}\:}$

$:\implies\:\:\bf\orange{QR\:=\:\sqrt{41}\:}$

FOR PR ;-

☆ For P(5 , -3) & R(4 , 6),

$\red\checkmark\:\:\bf{PR\:=\:\sqrt{(4\:-\:5)^2\:+\:(6\:-\:(-3))^2}\:}$

$:\implies\:\:\bf{PR\:=\:\sqrt{(-1)^2\:+\:9^2}\:}$

$:\implies\:\:\bf{PR\:=\:\sqrt{1\:+\:81}\:}$

$:\implies\:\:\bf\green{PR\:=\:\sqrt{82}\:}$

☆ For P(5 , -3) & R(-4 , 6),

$\red\checkmark\:\:\bf{PR\:=\:\sqrt{(-4\:-\:5)^2\:+\:(6\:-\:(-3))^2}\:}$

$:\implies\:\:\bf{PR\:=\:\sqrt{(-9)^2\:+\:9^2}\:}$

$:\implies\:\:\bf{PR\:=\:\sqrt{81\:+\:81}\:}$

$:\implies\:\:\bf{PR\:=\:\sqrt{162}\:}$

$:\implies\:\:\bf{\color{peru}PR\:=\:9\sqrt{2}\:}$