If the tangent to the conic, y — 6 = x2 at (2, 10 ) touches the circle, x2 + y2 + 8x — 2y = k (for some fixed k ) at a point (α, β); then (α, β) is :

Question

If the tangent to the conic, y — 6 = x2 at (2, 10 ) touches the circle, x2 + y2 + 8x — 2y = k (for some fixed k ) at a point (α, β); then (α, β) is : (A) (-(6/17), (10/17)) (B) (-(8/17), (2/17)) (C) (-(4/17), (1/17)) (D) (-(7/17), (6/17))

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/2aa8045376ac95e8fdec4902c89d3a6b-.png" /> Equation tangent to y=x2+6 at (2,10) is ((y+10/2))=2 x+6 i.e. 4 x-y+2=0...(1) ∵(1) is tangent to circle x2+y2+8 x-2 y=k Also P(α, β) must be foot of perpendicular from C(-4,1) to 4 x-y+2=0 ⇒ (x+4/4)=(y-1/-1)=([4(-4)-1+2]/16+1) ⇒ x+4=(60/17), y-1=(-15/17) ⇒ x=(-8/17), y=(2/17) ⇒ (α, β) ≡((-8/17), (2/17))

Q. If the tangent to the conic, $y —6 = x^{2}$ at ( $2, 10$ ) touches the circle, $x^{2} + y^{2} + 8 x —2 y = k$ (for some fixed $k$ ) at a point $(α, β)$ ; then $(α, β)$ is :

$(- \frac{6}{17}, \frac{10}{17})$

$(- \frac{8}{17}, \frac{2}{17})$

$(- \frac{4}{17}, \frac{1}{17})$

$(- \frac{7}{17}, \frac{6}{17})$

Q. If the tangent to the conic, y—6=x2 at (2,10 ) touches the circle, x2+y2+8x—2y=k (for some fixed k ) at a point (α,β); then (α,β) is :

(−176​,1710​)

(−178​,172​)

(−174​,171​)

(−177​,176​)

Q. If the tangent to the conic, $y —6 = x^{2}$ at ( $2, 10$ ) touches the circle, $x^{2} + y^{2} + 8 x —2 y = k$ (for some fixed $k$ ) at a point $(α, β)$ ; then $(α, β)$ is :

$(- \frac{6}{17}, \frac{10}{17})$

$(- \frac{8}{17}, \frac{2}{17})$

$(- \frac{4}{17}, \frac{1}{17})$

$(- \frac{7}{17}, \frac{6}{17})$