Question

Find the point of intersection of tangents drawn at the points (i) the points (2,-4) and $(\frac{1}{2},-2)$ of the parabola $y^2 = 8x$ (ii) the points (-2,2) and (-8,-4) of parabola $y^2 = -2x$.

• A. (i) (1, -3) and (ii) (4, -1)
• B. (i) (1, 3) and (ii) (-4, 1)
• C. (i) (-1, -3) and (ii) (-4, -1)
• D. (i) (1, -3) and (ii) (-4, 1)

Answer 1

Answer: (A)

(i) (1, -3) and (ii) (4, -1)

Answer 2

Part (i): Parabola $y^2 = 8x$

Comparing with $y^2 = 4ax$, we get $4a = 8$, so $a = 2$. The general equation of the tangent to the parabola $y^2=4ax$ at a point $(x_1, y_1)$ is $yy_1 = 2a(x+x_1)$.

1. Tangent at $(2, -4)$: Using the tangent equation with $x_1=2, y_1=-4, a=2$: $y(-4) = 2(2)(x+2)$ $-4y = 4(x+2)$ $-y = x+2 \implies y = -x - 2$ (Equation 1)

2. Tangent at $(\frac{1}{2}, -2)$: Using the tangent equation with $x_1=\frac{1}{2}, y_1=-2, a=2$: $y(-2) = 2(2)(x+\frac{1}{2})$ $-2y = 4(x+\frac{1}{2})$ $-y = 2(x+\frac{1}{2})$ $-y = 2x+1 \implies y = -2x - 1$ (Equation 2)

3. Point of Intersection: Equating Equation 1 and Equation 2: $-x - 2 = -2x - 1$ $2x - x = -1 + 2$ $x = 1$ Substitute $x=1$ into Equation 1: $y = -(1) - 2 = -3$ The point of intersection for part (i) is $(1, -3)$.

Part (ii): Parabola $y^2 = -2x$

Comparing with $y^2 = 4ax$, we get $4a = -2$, so $a = -\frac{1}{2}$.

1. Tangent at $(-2, 2)$: Using the tangent equation with $x_1=-2, y_1=2, a=-\frac{1}{2}$: $y(2) = 2(-\frac{1}{2})(x+(-2))$ $2y = -1(x-2)$ $2y = -x+2 \implies y = -\frac{1}{2}x + 1$ (Equation 3)

2. Tangent at $(-8, -4)$: Using the tangent equation with $x_1=-8, y_1=-4, a=-\frac{1}{2}$: $y(-4) = 2(-\frac{1}{2})(x+(-8))$ $-4y = -1(x-8)$ $-4y = -x+8 \implies y = \frac{1}{4}x - 2$ (Equation 4)

3. Point of Intersection: Equating Equation 3 and Equation 4: $-\frac{1}{2}x + 1 = \frac{1}{4}x - 2$ Multiply by 4 to clear denominators: $-2x + 4 = x - 8$ $12 = 3x$ $x = 4$ Substitute $x=4$ into Equation 3: $y = -\frac{1}{2}(4) + 1 = -2 + 1 = -1$ The point of intersection for part (ii) is $(4, -1)$.

Summary: (i) The point of intersection is $(1, -3)$. (ii) The point of intersection is $(4, -1)$.