A common tangent is drawn to the circle x2 + y2 = c2 and the parabola y2 = 4ax. If the angle which this tangent makes with the axis of x is (π/4) then the relationship between a and c is (a, c 0)

Question

A common tangent is drawn to the circle x2 + y2 = c2 and the parabola y2 = 4ax. If the angle which this tangent makes with the axis of x is (π/4) then the relationship between a and c is (a, c > 0) (A) a = √ 2C (B) c = a √ 2 (C) a = 2c (D) c = 2a

Tardigrade Admin · Accepted Answer

T is Y= mx+(a/m) ⇒ m2x - my +a=0 ∴|(a/√m4+m2)|=c a 2 = c2 (m4 + m2 ) (m = tan 45°) a 2 = 2c2 ⇒ a = √2 c

Q. A common tangent is drawn to the circle $x^{2} + y^{2} = c^{2}$ and the parabola $y^{2} = 4 a x$ . If the angle which this tangent makes with the axis of $x$ is $\frac{π}{4}$ then the relationship between a and c is $(a, c > 0)$

$a = 2 C$

$c = a 2$

$a = 2 c$

$c = 2 a$

$T$ is $Y = m x + \frac{a}{m} \Rightarrow m^{2} x - m y + a = 0$
$∴ ∣ ∣ \frac{a}{m ^{4} + m ^{2}} ∣ ∣ = c$
$a^{2} = c^{2} (m^{4} + m^{2}) (m = t an 4 5^{\circ})$
$a^{2} = 2 c^{2} \Rightarrow a = 2 c$

Q. A common tangent is drawn to the circle x2+y2=c2 and the parabola y2=4ax. If the angle which this tangent makes with the axis of x is 4π​ then the relationship between a and c is (a,c>0)

a=2​C

c=a2​

a=2c

c=2a