Question

A circle passes through the points of intersection of the parabola y + 1 = (x - 4)² and x-axis. Then the length of tangent from origin to the circle is:

• A. 8
• B. 15
• C. $\sqrt{8}$
• D. $\sqrt{15}$

Answer 1

Answer: (D)

$\sqrt{15}$

Answer 2

1. Find the points of intersection: The parabola is given by $y + 1 = (x - 4)^2$. The x-axis is given by $y = 0$. To find the points of intersection, substitute $y=0$ into the parabola's equation: $0 + 1 = (x - 4)^2$ $1 = (x - 4)^2$ Taking the square root of both sides: $x - 4 = \pm 1$ This gives two possible values for $x$: $x - 4 = 1 \implies x = 5$ $x - 4 = -1 \implies x = 3$ So, the points of intersection are $(3, 0)$ and $(5, 0)$.

2. Determine the general equation of a circle passing through these points: Let the general equation of a circle be $x^2 + y^2 + 2gx + 2fy + c = 0$. Since the circle passes through $(3, 0)$: $3^2 + 0^2 + 2g(3) + 2f(0) + c = 0$ $9 + 6g + c = 0 \quad (1)$

Since the circle passes through $(5, 0)$: $5^2 + 0^2 + 2g(5) + 2f(0) + c = 0$ $25 + 10g + c = 0 \quad (2)$

Now, we solve the system of linear equations (1) and (2) for $g$ and $c$. Subtract equation (1) from equation (2): $(25 + 10g + c) - (9 + 6g + c) = 0$ $16 + 4g = 0$ $4g = -16$ $g = -4$

Substitute the value of $g$ back into equation (1): $9 + 6(-4) + c = 0$ $9 - 24 + c = 0$ $-15 + c = 0$ $c = 15$

So, the equation of any circle passing through the points $(3,0)$ and $(5,0)$ must be of the form: $x^2 + y^2 - 8x + 2fy + 15 = 0$ Note that the value of $f$ determines the specific circle, but it does not affect the values of $g$ and $c$.

3. Calculate the length of the tangent from the origin: The length of the tangent from an external point $(x_0, y_0)$ to a circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is given by the formula $\sqrt{x_0^2 + y_0^2 + 2gx_0 + 2fy_0 + c}$. In this case, the external point is the origin $(0, 0)$. The equation of the circle is $x^2 + y^2 - 8x + 2fy + 15 = 0$. Here, $x_0 = 0$, $y_0 = 0$, $g = -4$, $f$ is some value, and $c = 15$.

Length of tangent = $\sqrt{0^2 + 0^2 + 2(-4)(0) + 2f(0) + 15}$ Length of tangent = $\sqrt{0 + 0 + 0 + 0 + 15}$ Length of tangent = $\sqrt{15}$

The length of the tangent from the origin to any circle passing through the points of intersection of the given parabola and the x-axis is $\sqrt{15}$.