Let A 1 , B 1 , C 1 be three points in the x y -plane. Suppose that the lines A 1 C 1 and B 1 C 1 are tangents to the curve y 2 = 8 x at A 1 and B 1 , respectively. If O = ( 0 , 0 ) and C 1 = ( - 4 , 0 ) , then which of the following statem

Question

Let $A_{1}, B_{1}, C_{1}$ be three points in the $x y$ -plane. Suppose that the lines $A_{1} C_{1}$ and $B_{1} C_{1}$ are tangents to the curve $y^{2} = 8 x$ at $A_{1}$ and $B_{1}$ , respectively. If $O = (0, 0)$ and $C_{1} = (- 4, 0)$ , then which of the following statements is (are) TRUE? [2024]

Smart Achievers · Accepted Answer

(3, 4)

Let parametric coordinates of $A_{1}$ and $B_{1}$ are

$A_{1} = (2 t_{1}^{2}, 4 t_{1}), B_{1} = (2 t_{2}^{2}, 4 t_{2})$

$C \equiv (- 4, 0) \equiv (2 t_{1} t_{2}, 2 (t_{1} + t_{2}))$

$\Rightarrow t_{2} = - t_{1} and t_{1} (- t_{1}) = - 2$

$t_{1} = \sqrt{2}, t_{2} = - \sqrt{2}$

$A_{1} \equiv (4, 4 \sqrt{2}), B_{1} \equiv (4, - 4 \sqrt{2})$

$∴$ $O A_{1} = \sqrt{4^{2} + {(4 \sqrt{2})}^{2}} = 4 \sqrt{3}$

$Length of line segment A_{1} B_{1} = 8 \sqrt{2}$

$Altitude C_{1} M : y = 0 . . . (i)$

$Altitude B_{1} N : \sqrt{2} x + y = 0 . . . (i i)$

$∴$ $Orthocentre \equiv (0, 0)$

Solution

Q.
Let $A_{1}, B_{1}, C_{1}$ be three points in the $x y$ -plane. Suppose that the lines $A_{1} C_{1}$ and $B_{1} C_{1}$ are tangents to the curve $y^{2} = 8 x$ at $A_{1}$ and $B_{1}$ , respectively. If $O = (0, 0)$ and $C_{1} = (- 4, 0)$ , then which of the following statements is (are) TRUE [2024]

1 The length of the line segment $O A_{1}$ is $4 \sqrt{3}$

2 The length of the line segment $A_{1} B_{1}$ is 16

3 The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(0, 0)$

4 The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(1, 0)$

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let A1,B1,C1 be three points in the xy-plane. Suppose that the lines A1C1 and B1C1 are tangents to the curve y2=8x at A1 and B1, respectively. If O=(0,0) and C1=(-4,0), then which of the following statements is (are) TRUE [2024]

1 The length of the line segment OA1 is 43

2 The length of the line segment A1B1 is 16

3 The orthocenter of the triangle A1B1C1 is (0,0)

4 The orthocenter of the triangle A1B1C1 is (1,0)

Want Us to Email you About Special Offers & Updates?

1 The length of the line segment $O A_{1}$ is $4 \sqrt{3}$

2 The length of the line segment $A_{1} B_{1}$ is 16

3 The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(0, 0)$

4 The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(1, 0)$