The focal chord of the parabola (y-2)2=16(x-1) is a tangent to the circle x2+y2-14 x-4 y+51=0, then slope of the focal chord can be

Question

The focal chord of the parabola (y-2)2=16(x-1) is a tangent to the circle x2+y2-14 x-4 y+51=0, then slope of the focal chord can be (A) 0 (B) 1 (C) 2 (D) 3

Tardigrade Admin · Accepted Answer

Focus of given parabola is (5,2). Now, any line through (5,2) is (y-2)=m(x-5) This will be a tangent to the circle (x-7)2+(y-2)2=2, if |(0-2 m/√1+ m 2)|=√2 ⇒ 4 m 2=2+2 m 2 ⇒ m = ± 1 .

Q. The focal chord of the parabola $(y - 2)^{2} = 16 (x - 1)$ is a tangent to the circle $x^{2} + y^{2} - 14 x - 4 y + 51 = 0,$ then slope of the focal chord can be

0

1

2

3

Focus of given parabola is $(5, 2)$ .
Now, any line through $(5, 2)$ is $(y - 2) = m (x - 5)$
This will be a tangent to the circle $(x - 7)^{2} + (y - 2)^{2} = 2$ ,
if $∣ ∣ \frac{0 - 2 m}{1 + m ^{2}} ∣ ∣ = 2 \Rightarrow 4 m^{2} = 2 + 2 m^{2} \Rightarrow m = \pm 1.$

Q. The focal chord of the parabola (y−2)2=16(x−1) is a tangent to the circle x2+y2−14x−4y+51=0, then slope of the focal chord can be

0

1

2

3

Focus of given parabola is (5,2). Now, any line through (5,2) is (y−2)=m(x−5) This will be a tangent to the circle (x−7)2+(y−2)2=2, if ∣∣​1+m2​0−2m​∣∣​=2​⇒4m2=2+2m2⇒m=±1.

Q. The focal chord of the parabola $(y - 2)^{2} = 16 (x - 1)$ is a tangent to the circle $x^{2} + y^{2} - 14 x - 4 y + 51 = 0,$ then slope of the focal chord can be

Focus of given parabola is $(5, 2)$ .
Now, any line through $(5, 2)$ is $(y - 2) = m (x - 5)$
This will be a tangent to the circle $(x - 7)^{2} + (y - 2)^{2} = 2$ ,
if $∣ ∣ \frac{0 - 2 m}{1 + m ^{2}} ∣ ∣ = 2 \Rightarrow 4 m^{2} = 2 + 2 m^{2} \Rightarrow m = \pm 1.$