Question

If the tangent to the parabola $y ^ { 2 } = a x$ makes an angle of 45 ^ { ullet } with x-axis, then the point of contact is

• A. $\left( \frac { a } { 2 } , \frac { a } { 2 } \right)$
• B. $\left( \frac { a } { 4 } , \frac { a } { 4 } \right)$
• C. $\left( \frac { a } { 2 } , \frac { a } { 4 } \right)$
• D. $\left( \frac { a } { 4 } , \frac { a } { 2 } \right)$

Answer 1

Answer: (D)

$\left( \frac { a } { 4 } , \frac { a } { 2 } \right)$

Answer 2

Parabola is $y ^ { 2 } = a x$ i.e. $y ^ { 2 } = 4 \left( \frac { a } { 4 } \right) x$ .....(i)

Let point of contact is (x₁, y₁). ∴ Equation of tangent is $y - y _ { 1 } = \frac { 2 ( a / 4 ) } { y _ { 1 } } \left( x - x _ { 1 } \right)$ ⇒ $y = \frac { a } { 2 y _ { 1 } } ( x ) - \frac { a x _ { 1 } } { 2 y _ { 1 } } + y _ { 1 }$

Here, $m = \frac { a } { 2 y _ { 1 } } = \tan 45 ^ { \circ }$ ⇒ $\frac { a } { 2 y _ { 1 } } = 1$ ⇒ $y _ { 1 } = \frac { a } { 2 }$ .

From (i), $x _ { 1 } = \frac { a } { 4 }$ So point is $\left( \frac { a } { 4 } , \frac { a } { 2 } \right)$