Question

A line passes through the point $(-1, 1)$ and makes an angle $\sin^{-1} \left( \frac{3}{5} \right)$ with the positive direction of the $x$-axis. If this line meets the curve $x^2 = 4y - 9$ at $A$ and $B$, then $|AB|$ is equal to

• A. 4/5 unit
• B. 5/4 unit
• C. 3/5 unit
• D. 5/3 unit

Answer 1

The Correct Option is B

Given: - A line passes through point \( (-1, 1) \) - It makes an angle \( \theta = \sin^{-1} \left( \frac{3}{5} \right) \) with the positive \( x \)-axis - The line intersects the curve \( x^2 = 4y - 9 \) at points \( A \) and \( B \) We are to find the length \( |AB| \) Step 1: Determine the slope of the line Given \( \theta = \sin^{-1} \left( \frac{3}{5} \right) \), we can construct a right triangle: - \( \sin \theta = \frac{3}{5} \) ⇒ opposite = 3, hypotenuse = 5 - So, adjacent = \( \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = 4 \) ⇒ \( \cos \theta = \frac{4}{5} \) Then: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{3/5}{4/5} = \frac{3}{4} \] So, the slope of the line is \( m = \frac{3}{4} \) Step 2: Find the equation of the line Using point-slope form with point \( (-1, 1) \) and slope \( \frac{3}{4} \): \[ y - 1 = \frac{3}{4}(x + 1) \Rightarrow y = \frac{3}{4}(x + 1) + 1 = \frac{3x}{4} + \frac{7}{4} \] Step 3: Find points of intersection with the parabola The curve is \( x^2 = 4y - 9 \Rightarrow y = \frac{x^2 + 9}{4} \) Now equate the expressions for \( y \): \[ \frac{3x}{4} + \frac{7}{4} = \frac{x^2 + 9}{4} \Rightarrow 3x + 7 = x^2 + 9 \Rightarrow x^2 - 3x + 2 = 0 \Rightarrow (x - 1)(x - 2) = 0 \Rightarrow x = 1 \text{ or } 2 \] Step 4: Find corresponding y-values Using the line equation \( y = \frac{3x}{4} + \frac{7}{4} \): - If \( x = 1 \), then \( y = \frac{3(1)}{4} + \frac{7}{4} = \frac{10}{4} = \frac{5}{2} \) - If \( x = 2 \), then \( y = \frac{3(2)}{4} + \frac{7}{4} = \frac{13}{4} \) So points \( A = (1, \frac{5}{2}) \), \( B = (2, \frac{13}{4}) \) Step 5: Find the distance \( |AB| \) Use distance formula: \[ |AB| = \sqrt{(2 - 1)^2 + \left( \frac{13}{4} - \frac{5}{2} \right)^2} = \sqrt{1^2 + \left( \frac{13 - 10}{4} \right)^2} = \sqrt{1 + \left( \frac{3}{4} \right)^2} = \sqrt{1 + \frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} \] Final Answer: \[ \boxed{ \frac{5}{4} \text{ unit} } \]